Theorem fpwwe2 10596

Description: Given any function 𝐹 from well-orderings of subsets of 𝐴 to 𝐴, there is a unique well-ordered subset ⟨𝑋, (𝑊‘𝑋)⟩ which "agrees" with 𝐹 in the sense that each initial segment maps to its upper bound, and such that the entire set maps to an element of the set (so that it cannot be extended without losing the well-ordering). This theorem can be used to prove dfac8a 9983. Theorem 1.1 of [KanamoriPincus] p. 415. (Contributed by Mario Carneiro, 18-May-2015.) (Revised by AV, 20-Jul-2024.)

Hypotheses

Ref	Expression
fpwwe2.1	⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
fpwwe2.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)
fpwwe2.3	⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
fpwwe2.4	⊢ 𝑋 = ∪ dom 𝑊

Assertion

Ref	Expression
fpwwe2	⊢ (𝜑 → ((𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌) ↔ (𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋))))

Distinct variable groups:   𝑦,𝑢,𝑟,𝑥,𝐹   𝑋,𝑟,𝑢,𝑥,𝑦   𝜑,𝑟,𝑢,𝑥,𝑦   𝐴,𝑟,𝑥   𝑅,𝑟,𝑢,𝑥,𝑦   𝑌,𝑟,𝑢,𝑥,𝑦   𝑊,𝑟,𝑢,𝑥,𝑦

Allowed substitution hints:   𝐴(𝑦,𝑢)   𝑉(𝑥,𝑦,𝑢,𝑟)

Proof of Theorem fpwwe2

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fpwwe2.1	. . . . . . . . . . 11 ⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
2		fpwwe2.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		fpwwe2.3	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
4		fpwwe2.4	. . . . . . . . . . 11 ⊢ 𝑋 = ∪ dom 𝑊
5	1, 2, 3, 4	fpwwe2lem10 10593	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊:dom 𝑊⟶𝒫 (𝑋 × 𝑋))
6	5	ffund 6692	. . . . . . . . 9 ⊢ (𝜑 → Fun 𝑊)
7		funbrfv2b 6918	. . . . . . . . 9 ⊢ (Fun 𝑊 → (𝑌𝑊𝑅 ↔ (𝑌 ∈ dom 𝑊 ∧ (𝑊‘𝑌) = 𝑅)))
8	6, 7	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑌𝑊𝑅 ↔ (𝑌 ∈ dom 𝑊 ∧ (𝑊‘𝑌) = 𝑅)))
9	8	simprbda 498	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌𝑊𝑅) → 𝑌 ∈ dom 𝑊)
10	9	adantrr 717	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑌 ∈ dom 𝑊)
11		elssuni 4901	. . . . . . 7 ⊢ (𝑌 ∈ dom 𝑊 → 𝑌 ⊆ ∪ dom 𝑊)
12	11, 4	sseqtrrdi 3988	. . . . . 6 ⊢ (𝑌 ∈ dom 𝑊 → 𝑌 ⊆ 𝑋)
13	10, 12	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑌 ⊆ 𝑋)
14		simpl 482	. . . . . . 7 ⊢ ((𝑋 ⊆ 𝑌 ∧ (𝑊‘𝑋) = (𝑅 ∩ (𝑌 × 𝑋))) → 𝑋 ⊆ 𝑌)
15	14	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → ((𝑋 ⊆ 𝑌 ∧ (𝑊‘𝑋) = (𝑅 ∩ (𝑌 × 𝑋))) → 𝑋 ⊆ 𝑌))
16		simplrr 777	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (𝑌𝐹𝑅) ∈ 𝑌)
17	2	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝐴 ∈ 𝑉)
18	17	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → 𝐴 ∈ 𝑉)
19	1, 2, 3, 4	fpwwe2lem11 10594	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑋 ∈ dom 𝑊)
20		funfvbrb 7023	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Fun 𝑊 → (𝑋 ∈ dom 𝑊 ↔ 𝑋𝑊(𝑊‘𝑋)))
21	6, 20	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑋 ∈ dom 𝑊 ↔ 𝑋𝑊(𝑊‘𝑋)))
22	19, 21	mpbid 232	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑋𝑊(𝑊‘𝑋))
23	1, 2	fpwwe2lem2 10585	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑋𝑊(𝑊‘𝑋) ↔ ((𝑋 ⊆ 𝐴 ∧ (𝑊‘𝑋) ⊆ (𝑋 × 𝑋)) ∧ ((𝑊‘𝑋) We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡(𝑊‘𝑋) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊‘𝑋) ∩ (𝑢 × 𝑢))) = 𝑦))))
24	22, 23	mpbid 232	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑋 ⊆ 𝐴 ∧ (𝑊‘𝑋) ⊆ (𝑋 × 𝑋)) ∧ ((𝑊‘𝑋) We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡(𝑊‘𝑋) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊‘𝑋) ∩ (𝑢 × 𝑢))) = 𝑦)))
25	24	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑋 ⊆ 𝐴 ∧ (𝑊‘𝑋) ⊆ (𝑋 × 𝑋)) ∧ ((𝑊‘𝑋) We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡(𝑊‘𝑋) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊‘𝑋) ∩ (𝑢 × 𝑢))) = 𝑦)))
26	25	simpld 494	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (𝑋 ⊆ 𝐴 ∧ (𝑊‘𝑋) ⊆ (𝑋 × 𝑋)))
27	26	simpld 494	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → 𝑋 ⊆ 𝐴)
28	18, 27	ssexd 5279	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → 𝑋 ∈ V)
29	28	difexd 5286	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (𝑋 ∖ 𝑌) ∈ V)
30	25	simprd 495	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑊‘𝑋) We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡(𝑊‘𝑋) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊‘𝑋) ∩ (𝑢 × 𝑢))) = 𝑦))
31	30	simpld 494	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (𝑊‘𝑋) We 𝑋)
32		wefr 5628	. . . . . . . . . . . . 13 ⊢ ((𝑊‘𝑋) We 𝑋 → (𝑊‘𝑋) Fr 𝑋)
33	31, 32	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (𝑊‘𝑋) Fr 𝑋)
34		difssd 4100	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (𝑋 ∖ 𝑌) ⊆ 𝑋)
35		fri 5596	. . . . . . . . . . . . 13 ⊢ ((((𝑋 ∖ 𝑌) ∈ V ∧ (𝑊‘𝑋) Fr 𝑋) ∧ ((𝑋 ∖ 𝑌) ⊆ 𝑋 ∧ (𝑋 ∖ 𝑌) ≠ ∅)) → ∃𝑧 ∈ (𝑋 ∖ 𝑌)∀𝑤 ∈ (𝑋 ∖ 𝑌) ¬ 𝑤(𝑊‘𝑋)𝑧)
36	35	expr 456	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∖ 𝑌) ∈ V ∧ (𝑊‘𝑋) Fr 𝑋) ∧ (𝑋 ∖ 𝑌) ⊆ 𝑋) → ((𝑋 ∖ 𝑌) ≠ ∅ → ∃𝑧 ∈ (𝑋 ∖ 𝑌)∀𝑤 ∈ (𝑋 ∖ 𝑌) ¬ 𝑤(𝑊‘𝑋)𝑧))
37	29, 33, 34, 36	syl21anc 837	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑋 ∖ 𝑌) ≠ ∅ → ∃𝑧 ∈ (𝑋 ∖ 𝑌)∀𝑤 ∈ (𝑋 ∖ 𝑌) ¬ 𝑤(𝑊‘𝑋)𝑧))
38		ssdif0 4329	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∩ (◡(𝑊‘𝑋) “ {𝑧})) ⊆ 𝑌 ↔ ((𝑋 ∩ (◡(𝑊‘𝑋) “ {𝑧})) ∖ 𝑌) = ∅)
39		indif1 4245	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∖ 𝑌) ∩ (◡(𝑊‘𝑋) “ {𝑧})) = ((𝑋 ∩ (◡(𝑊‘𝑋) “ {𝑧})) ∖ 𝑌)
40	39	eqeq1i 2734	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∖ 𝑌) ∩ (◡(𝑊‘𝑋) “ {𝑧})) = ∅ ↔ ((𝑋 ∩ (◡(𝑊‘𝑋) “ {𝑧})) ∖ 𝑌) = ∅)
41		disj 4413	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∖ 𝑌) ∩ (◡(𝑊‘𝑋) “ {𝑧})) = ∅ ↔ ∀𝑤 ∈ (𝑋 ∖ 𝑌) ¬ 𝑤 ∈ (◡(𝑊‘𝑋) “ {𝑧}))
42		vex 3451	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑤 ∈ V
43	42	eliniseg 6065	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ V → (𝑤 ∈ (◡(𝑊‘𝑋) “ {𝑧}) ↔ 𝑤(𝑊‘𝑋)𝑧))
44	43	elv 3452	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ (◡(𝑊‘𝑋) “ {𝑧}) ↔ 𝑤(𝑊‘𝑋)𝑧)
45	44	notbii 320	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑤 ∈ (◡(𝑊‘𝑋) “ {𝑧}) ↔ ¬ 𝑤(𝑊‘𝑋)𝑧)
46	45	ralbii 3075	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑤 ∈ (𝑋 ∖ 𝑌) ¬ 𝑤 ∈ (◡(𝑊‘𝑋) “ {𝑧}) ↔ ∀𝑤 ∈ (𝑋 ∖ 𝑌) ¬ 𝑤(𝑊‘𝑋)𝑧)
47	41, 46	bitri 275	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∖ 𝑌) ∩ (◡(𝑊‘𝑋) “ {𝑧})) = ∅ ↔ ∀𝑤 ∈ (𝑋 ∖ 𝑌) ¬ 𝑤(𝑊‘𝑋)𝑧)
48	38, 40, 47	3bitr2i 299	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∩ (◡(𝑊‘𝑋) “ {𝑧})) ⊆ 𝑌 ↔ ∀𝑤 ∈ (𝑋 ∖ 𝑌) ¬ 𝑤(𝑊‘𝑋)𝑧)
49		cnvimass 6053	. . . . . . . . . . . . . . . . 17 ⊢ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ dom (𝑊‘𝑋)
50	26	simprd 495	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (𝑊‘𝑋) ⊆ (𝑋 × 𝑋))
51		dmss 5866	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑊‘𝑋) ⊆ (𝑋 × 𝑋) → dom (𝑊‘𝑋) ⊆ dom (𝑋 × 𝑋))
52	50, 51	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → dom (𝑊‘𝑋) ⊆ dom (𝑋 × 𝑋))
53		dmxpid 5894	. . . . . . . . . . . . . . . . . 18 ⊢ dom (𝑋 × 𝑋) = 𝑋
54	52, 53	sseqtrdi 3987	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → dom (𝑊‘𝑋) ⊆ 𝑋)
55	49, 54	sstrid 3958	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑋)
56		sseqin2 4186	. . . . . . . . . . . . . . . 16 ⊢ ((◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑋 ↔ (𝑋 ∩ (◡(𝑊‘𝑋) “ {𝑧})) = (◡(𝑊‘𝑋) “ {𝑧}))
57	55, 56	sylib 218	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (𝑋 ∩ (◡(𝑊‘𝑋) “ {𝑧})) = (◡(𝑊‘𝑋) “ {𝑧}))
58	57	sseq1d 3978	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑋 ∩ (◡(𝑊‘𝑋) “ {𝑧})) ⊆ 𝑌 ↔ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌))
59	48, 58	bitr3id 285	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (∀𝑤 ∈ (𝑋 ∖ 𝑌) ¬ 𝑤(𝑊‘𝑋)𝑧 ↔ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌))
60	59	rexbidv 3157	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (∃𝑧 ∈ (𝑋 ∖ 𝑌)∀𝑤 ∈ (𝑋 ∖ 𝑌) ¬ 𝑤(𝑊‘𝑋)𝑧 ↔ ∃𝑧 ∈ (𝑋 ∖ 𝑌)(◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌))
61		eldifn 4095	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑧 ∈ (𝑋 ∖ 𝑌) → ¬ 𝑧 ∈ 𝑌)
62	61	ad2antrl 728	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → ¬ 𝑧 ∈ 𝑌)
63		eleq1w 2811	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ 𝑌 ↔ 𝑧 ∈ 𝑌))
64	63	notbid 318	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = 𝑧 → (¬ 𝑤 ∈ 𝑌 ↔ ¬ 𝑧 ∈ 𝑌))
65	62, 64	syl5ibrcom 247	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑤 = 𝑧 → ¬ 𝑤 ∈ 𝑌))
66	65	con2d 134	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑤 ∈ 𝑌 → ¬ 𝑤 = 𝑧))
67	66	imp 406	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → ¬ 𝑤 = 𝑧)
68	62	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → ¬ 𝑧 ∈ 𝑌)
69		simprr 772	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))
70	69	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))
71	70	breqd 5118	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → (𝑧𝑅𝑤 ↔ 𝑧((𝑊‘𝑋) ∩ (𝑋 × 𝑌))𝑤))
72		eldifi 4094	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 ∈ (𝑋 ∖ 𝑌) → 𝑧 ∈ 𝑋)
73	72	ad2antrl 728	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑧 ∈ 𝑋)
74	73	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → 𝑧 ∈ 𝑋)
75		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → 𝑤 ∈ 𝑌)
76		brxp 5687	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧(𝑋 × 𝑌)𝑤 ↔ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑌))
77	74, 75, 76	sylanbrc 583	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → 𝑧(𝑋 × 𝑌)𝑤)
78		brin 5159	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧((𝑊‘𝑋) ∩ (𝑋 × 𝑌))𝑤 ↔ (𝑧(𝑊‘𝑋)𝑤 ∧ 𝑧(𝑋 × 𝑌)𝑤))
79	78	rbaib 538	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑧(𝑋 × 𝑌)𝑤 → (𝑧((𝑊‘𝑋) ∩ (𝑋 × 𝑌))𝑤 ↔ 𝑧(𝑊‘𝑋)𝑤))
80	77, 79	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → (𝑧((𝑊‘𝑋) ∩ (𝑋 × 𝑌))𝑤 ↔ 𝑧(𝑊‘𝑋)𝑤))
81	71, 80	bitrd 279	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → (𝑧𝑅𝑤 ↔ 𝑧(𝑊‘𝑋)𝑤))
82	1, 2	fpwwe2lem2 10585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → (𝑌𝑊𝑅 ↔ ((𝑌 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑌 × 𝑌)) ∧ (𝑅 We 𝑌 ∧ ∀𝑦 ∈ 𝑌 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))))
83	82	biimpa 476	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑌𝑊𝑅) → ((𝑌 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑌 × 𝑌)) ∧ (𝑅 We 𝑌 ∧ ∀𝑦 ∈ 𝑌 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦)))
84	83	adantrr 717	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → ((𝑌 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑌 × 𝑌)) ∧ (𝑅 We 𝑌 ∧ ∀𝑦 ∈ 𝑌 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦)))
85	84	simpld 494	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → (𝑌 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑌 × 𝑌)))
86	85	simprd 495	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑅 ⊆ (𝑌 × 𝑌))
87	86	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → 𝑅 ⊆ (𝑌 × 𝑌))
88	87	ssbrd 5150	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → (𝑧𝑅𝑤 → 𝑧(𝑌 × 𝑌)𝑤))
89		brxp 5687	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑧(𝑌 × 𝑌)𝑤 ↔ (𝑧 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌))
90	89	simplbi 497	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑧(𝑌 × 𝑌)𝑤 → 𝑧 ∈ 𝑌)
91	88, 90	syl6 35	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → (𝑧𝑅𝑤 → 𝑧 ∈ 𝑌))
92	81, 91	sylbird 260	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → (𝑧(𝑊‘𝑋)𝑤 → 𝑧 ∈ 𝑌))
93	68, 92	mtod 198	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → ¬ 𝑧(𝑊‘𝑋)𝑤)
94	31	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → (𝑊‘𝑋) We 𝑋)
95		weso 5629	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑊‘𝑋) We 𝑋 → (𝑊‘𝑋) Or 𝑋)
96	94, 95	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → (𝑊‘𝑋) Or 𝑋)
97	13	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑌 ⊆ 𝑋)
98	97	sselda 3946	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → 𝑤 ∈ 𝑋)
99		sotric 5576	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑊‘𝑋) Or 𝑋 ∧ (𝑤 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑤(𝑊‘𝑋)𝑧 ↔ ¬ (𝑤 = 𝑧 ∨ 𝑧(𝑊‘𝑋)𝑤)))
100		ioran 985	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (¬ (𝑤 = 𝑧 ∨ 𝑧(𝑊‘𝑋)𝑤) ↔ (¬ 𝑤 = 𝑧 ∧ ¬ 𝑧(𝑊‘𝑋)𝑤))
101	99, 100	bitrdi 287	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑊‘𝑋) Or 𝑋 ∧ (𝑤 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑤(𝑊‘𝑋)𝑧 ↔ (¬ 𝑤 = 𝑧 ∧ ¬ 𝑧(𝑊‘𝑋)𝑤)))
102	96, 98, 74, 101	syl12anc 836	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → (𝑤(𝑊‘𝑋)𝑧 ↔ (¬ 𝑤 = 𝑧 ∧ ¬ 𝑧(𝑊‘𝑋)𝑤)))
103	67, 93, 102	mpbir2and 713	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → 𝑤(𝑊‘𝑋)𝑧)
104	103, 44	sylibr 234	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → 𝑤 ∈ (◡(𝑊‘𝑋) “ {𝑧}))
105	104	ex 412	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑤 ∈ 𝑌 → 𝑤 ∈ (◡(𝑊‘𝑋) “ {𝑧})))
106	105	ssrdv 3952	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑌 ⊆ (◡(𝑊‘𝑋) “ {𝑧}))
107		simprr 772	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)
108	106, 107	eqssd 3964	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑌 = (◡(𝑊‘𝑋) “ {𝑧}))
109		in32 4193	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑊‘𝑋) ∩ (𝑋 × 𝑌)) ∩ (𝑌 × 𝑌)) = (((𝑊‘𝑋) ∩ (𝑌 × 𝑌)) ∩ (𝑋 × 𝑌))
110		simplrr 777	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))
111	110	ineq1d 4182	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑅 ∩ (𝑌 × 𝑌)) = (((𝑊‘𝑋) ∩ (𝑋 × 𝑌)) ∩ (𝑌 × 𝑌)))
112	86	ad2antrr 726	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑅 ⊆ (𝑌 × 𝑌))
113		dfss2 3932	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑅 ⊆ (𝑌 × 𝑌) ↔ (𝑅 ∩ (𝑌 × 𝑌)) = 𝑅)
114	112, 113	sylib 218	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑅 ∩ (𝑌 × 𝑌)) = 𝑅)
115	111, 114	eqtr3d 2766	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → (((𝑊‘𝑋) ∩ (𝑋 × 𝑌)) ∩ (𝑌 × 𝑌)) = 𝑅)
116		inss2 4201	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑊‘𝑋) ∩ (𝑌 × 𝑌)) ⊆ (𝑌 × 𝑌)
117		xpss1 5657	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑌 ⊆ 𝑋 → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑌))
118	97, 117	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑌))
119	116, 118	sstrid 3958	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → ((𝑊‘𝑋) ∩ (𝑌 × 𝑌)) ⊆ (𝑋 × 𝑌))
120		dfss2 3932	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑊‘𝑋) ∩ (𝑌 × 𝑌)) ⊆ (𝑋 × 𝑌) ↔ (((𝑊‘𝑋) ∩ (𝑌 × 𝑌)) ∩ (𝑋 × 𝑌)) = ((𝑊‘𝑋) ∩ (𝑌 × 𝑌)))
121	119, 120	sylib 218	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → (((𝑊‘𝑋) ∩ (𝑌 × 𝑌)) ∩ (𝑋 × 𝑌)) = ((𝑊‘𝑋) ∩ (𝑌 × 𝑌)))
122	109, 115, 121	3eqtr3a 2788	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑅 = ((𝑊‘𝑋) ∩ (𝑌 × 𝑌)))
123	108	sqxpeqd 5670	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑌 × 𝑌) = ((◡(𝑊‘𝑋) “ {𝑧}) × (◡(𝑊‘𝑋) “ {𝑧})))
124	123	ineq2d 4183	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → ((𝑊‘𝑋) ∩ (𝑌 × 𝑌)) = ((𝑊‘𝑋) ∩ ((◡(𝑊‘𝑋) “ {𝑧}) × (◡(𝑊‘𝑋) “ {𝑧}))))
125	122, 124	eqtrd 2764	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑅 = ((𝑊‘𝑋) ∩ ((◡(𝑊‘𝑋) “ {𝑧}) × (◡(𝑊‘𝑋) “ {𝑧}))))
126	108, 125	oveq12d 7405	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑌𝐹𝑅) = ((◡(𝑊‘𝑋) “ {𝑧})𝐹((𝑊‘𝑋) ∩ ((◡(𝑊‘𝑋) “ {𝑧}) × (◡(𝑊‘𝑋) “ {𝑧})))))
127	18	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝐴 ∈ 𝑉)
128	22	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑋𝑊(𝑊‘𝑋))
129	128	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑋𝑊(𝑊‘𝑋))
130	1, 127, 129	fpwwe2lem3 10586	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑧 ∈ 𝑋) → ((◡(𝑊‘𝑋) “ {𝑧})𝐹((𝑊‘𝑋) ∩ ((◡(𝑊‘𝑋) “ {𝑧}) × (◡(𝑊‘𝑋) “ {𝑧})))) = 𝑧)
131	73, 130	mpdan 687	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → ((◡(𝑊‘𝑋) “ {𝑧})𝐹((𝑊‘𝑋) ∩ ((◡(𝑊‘𝑋) “ {𝑧}) × (◡(𝑊‘𝑋) “ {𝑧})))) = 𝑧)
132	126, 131	eqtrd 2764	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑌𝐹𝑅) = 𝑧)
133	132, 62	eqneltrd 2848	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → ¬ (𝑌𝐹𝑅) ∈ 𝑌)
134	133	rexlimdvaa 3135	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (∃𝑧 ∈ (𝑋 ∖ 𝑌)(◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌 → ¬ (𝑌𝐹𝑅) ∈ 𝑌))
135	60, 134	sylbid 240	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (∃𝑧 ∈ (𝑋 ∖ 𝑌)∀𝑤 ∈ (𝑋 ∖ 𝑌) ¬ 𝑤(𝑊‘𝑋)𝑧 → ¬ (𝑌𝐹𝑅) ∈ 𝑌))
136	37, 135	syld 47	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑋 ∖ 𝑌) ≠ ∅ → ¬ (𝑌𝐹𝑅) ∈ 𝑌))
137	136	necon4ad 2944	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑌𝐹𝑅) ∈ 𝑌 → (𝑋 ∖ 𝑌) = ∅))
138	16, 137	mpd 15	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (𝑋 ∖ 𝑌) = ∅)
139		ssdif0 4329	. . . . . . . 8 ⊢ (𝑋 ⊆ 𝑌 ↔ (𝑋 ∖ 𝑌) = ∅)
140	138, 139	sylibr 234	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → 𝑋 ⊆ 𝑌)
141	140	ex 412	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → ((𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌))) → 𝑋 ⊆ 𝑌))
142	3	adantlr 715	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
143		simprl 770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑌𝑊𝑅)
144	1, 17, 142, 128, 143	fpwwe2lem9 10592	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → ((𝑋 ⊆ 𝑌 ∧ (𝑊‘𝑋) = (𝑅 ∩ (𝑌 × 𝑋))) ∨ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))))
145	15, 141, 144	mpjaod 860	. . . . 5 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑋 ⊆ 𝑌)
146	13, 145	eqssd 3964	. . . 4 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑌 = 𝑋)
147	6	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → Fun 𝑊)
148	146, 143	eqbrtrrd 5131	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑋𝑊𝑅)
149		funbrfv 6909	. . . . . 6 ⊢ (Fun 𝑊 → (𝑋𝑊𝑅 → (𝑊‘𝑋) = 𝑅))
150	147, 148, 149	sylc 65	. . . . 5 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → (𝑊‘𝑋) = 𝑅)
151	150	eqcomd 2735	. . . 4 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑅 = (𝑊‘𝑋))
152	146, 151	jca 511	. . 3 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → (𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋)))
153	152	ex 412	. 2 ⊢ (𝜑 → ((𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌) → (𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋))))
154	1, 2, 3, 4	fpwwe2lem12 10595	. . . 4 ⊢ (𝜑 → (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋)
155	22, 154	jca 511	. . 3 ⊢ (𝜑 → (𝑋𝑊(𝑊‘𝑋) ∧ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋))
156		breq12 5112	. . . 4 ⊢ ((𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋)) → (𝑌𝑊𝑅 ↔ 𝑋𝑊(𝑊‘𝑋)))
157		oveq12 7396	. . . . 5 ⊢ ((𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋)) → (𝑌𝐹𝑅) = (𝑋𝐹(𝑊‘𝑋)))
158		simpl 482	. . . . 5 ⊢ ((𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋)) → 𝑌 = 𝑋)
159	157, 158	eleq12d 2822	. . . 4 ⊢ ((𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋)) → ((𝑌𝐹𝑅) ∈ 𝑌 ↔ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋))
160	156, 159	anbi12d 632	. . 3 ⊢ ((𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋)) → ((𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌) ↔ (𝑋𝑊(𝑊‘𝑋) ∧ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋)))
161	155, 160	syl5ibrcom 247	. 2 ⊢ (𝜑 → ((𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋)) → (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)))
162	153, 161	impbid 212	1 ⊢ (𝜑 → ((𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌) ↔ (𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  Vcvv 3447  [wsbc 3753   ∖ cdif 3911   ∩ cin 3913   ⊆ wss 3914  ∅c0 4296  𝒫 cpw 4563  {csn 4589  ∪ cuni 4871   class class class wbr 5107  {copab 5169   Or wor 5545   Fr wfr 5588   We wwe 5590   × cxp 5636  ^◡ccnv 5637  dom cdm 5638   “ cima 5641  Fun wfun 6505  ‘cfv 6511  (class class class)co 7387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-oi 9463

This theorem is referenced by:  fpwwe  10599  canthwelem  10603  pwfseqlem4  10615