Theorem fpwwe2 10554

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 9940. 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 10551	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊:dom 𝑊⟶𝒫 (𝑋 × 𝑋))
6	5	ffund 6666	. . . . . . . . 9 ⊢ (𝜑 → Fun 𝑊)
7		funbrfv2b 6891	. . . . . . . . 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 4894	. . . . . . 7 ⊢ (𝑌 ∈ dom 𝑊 → 𝑌 ⊆ ∪ dom 𝑊)
12	11, 4	sseqtrrdi 3975	. . . . . 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 10552	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑋 ∈ dom 𝑊)
20		funfvbrb 6996	. . . . . . . . . . . . . . . . . . . 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 10543	. . . . . . . . . . . . . . . . . 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 5269	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → 𝑋 ∈ V)
29	28	difexd 5276	. . . . . . . . . . . 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 5614	. . . . . . . . . . . . 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 4089	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (𝑋 ∖ 𝑌) ⊆ 𝑋)
35		fri 5582	. . . . . . . . . . . . 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 4318	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∩ (◡(𝑊‘𝑋) “ {𝑧})) ⊆ 𝑌 ↔ ((𝑋 ∩ (◡(𝑊‘𝑋) “ {𝑧})) ∖ 𝑌) = ∅)
39		indif1 4234	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∖ 𝑌) ∩ (◡(𝑊‘𝑋) “ {𝑧})) = ((𝑋 ∩ (◡(𝑊‘𝑋) “ {𝑧})) ∖ 𝑌)
40	39	eqeq1i 2741	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∖ 𝑌) ∩ (◡(𝑊‘𝑋) “ {𝑧})) = ∅ ↔ ((𝑋 ∩ (◡(𝑊‘𝑋) “ {𝑧})) ∖ 𝑌) = ∅)
41		disj 4402	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∖ 𝑌) ∩ (◡(𝑊‘𝑋) “ {𝑧})) = ∅ ↔ ∀𝑤 ∈ (𝑋 ∖ 𝑌) ¬ 𝑤 ∈ (◡(𝑊‘𝑋) “ {𝑧}))
42		vex 3444	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑤 ∈ V
43	42	eliniseg 6053	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ V → (𝑤 ∈ (◡(𝑊‘𝑋) “ {𝑧}) ↔ 𝑤(𝑊‘𝑋)𝑧))
44	43	elv 3445	. . . . . . . . . . . . . . . . . 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 3082	. . . . . . . . . . . . . . . 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 6041	. . . . . . . . . . . . . . . . 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 5851	. . . . . . . . . . . . . . . . . . 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 5879	. . . . . . . . . . . . . . . . . 18 ⊢ dom (𝑋 × 𝑋) = 𝑋
54	52, 53	sseqtrdi 3974	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → dom (𝑊‘𝑋) ⊆ 𝑋)
55	49, 54	sstrid 3945	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑋)
56		sseqin2 4175	. . . . . . . . . . . . . . . 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 3965	. . . . . . . . . . . . . 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 3160	. . . . . . . . . . . 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 4084	. . . . . . . . . . . . . . . . . . . . . . . . 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 2819	. . . . . . . . . . . . . . . . . . . . . . . . 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 5109	. . . . . . . . . . . . . . . . . . . . . . . 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 4083	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5673	. . . . . . . . . . . . . . . . . . . . . . . . . 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 5150	. . . . . . . . . . . . . . . . . . . . . . . . . 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 10543	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5141	. . . . . . . . . . . . . . . . . . . . . . . 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 5673	. . . . . . . . . . . . . . . . . . . . . . . . 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 5615	. . . . . . . . . . . . . . . . . . . . . . 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 3933	. . . . . . . . . . . . . . . . . . . . . 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 5562	. . . . . . . . . . . . . . . . . . . . . . 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 3939	. . . . . . . . . . . . . . . . 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 3951	. . . . . . . . . . . . . . . 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 4182	. . . . . . . . . . . . . . . . . 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 4171	. . . . . . . . . . . . . . . . . . 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 3919	. . . . . . . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . . . . . 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 4190	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑊‘𝑋) ∩ (𝑌 × 𝑌)) ⊆ (𝑌 × 𝑌)
117		xpss1 5643	. . . . . . . . . . . . . . . . . . . . 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 3945	. . . . . . . . . . . . . . . . . . 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 3919	. . . . . . . . . . . . . . . . . . 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 2795	. . . . . . . . . . . . . . . . 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 5656	. . . . . . . . . . . . . . . . . 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 4172	. . . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . . 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 7376	. . . . . . . . . . . . . . 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 10544	. . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . 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 2856	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → ¬ (𝑌𝐹𝑅) ∈ 𝑌)
134	133	rexlimdvaa 3138	. . . . . . . . . . . 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 2951	. . . . . . . . 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 4318	. . . . . . . 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 10550	. . . . . 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 3951	. . . 4 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑌 = 𝑋)
147	6	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → Fun 𝑊)
148	146, 143	eqbrtrrd 5122	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑋𝑊𝑅)
149		funbrfv 6882	. . . . . 6 ⊢ (Fun 𝑊 → (𝑋𝑊𝑅 → (𝑊‘𝑋) = 𝑅))
150	147, 148, 149	sylc 65	. . . . 5 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → (𝑊‘𝑋) = 𝑅)
151	150	eqcomd 2742	. . . 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 10553	. . . 4 ⊢ (𝜑 → (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋)
155	22, 154	jca 511	. . 3 ⊢ (𝜑 → (𝑋𝑊(𝑊‘𝑋) ∧ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋))
156		breq12 5103	. . . 4 ⊢ ((𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋)) → (𝑌𝑊𝑅 ↔ 𝑋𝑊(𝑊‘𝑋)))
157		oveq12 7367	. . . . 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 2830	. . . 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 1541   ∈ wcel 2113   ≠ wne 2932  ∀wral 3051  ∃wrex 3060  Vcvv 3440  [wsbc 3740   ∖ cdif 3898   ∩ cin 3900   ⊆ wss 3901  ∅c0 4285  𝒫 cpw 4554  {csn 4580  ∪ cuni 4863   class class class wbr 5098  {copab 5160   Or wor 5531   Fr wfr 5574   We wwe 5576   × cxp 5622  ^◡ccnv 5623  dom cdm 5624   “ cima 5627  Fun wfun 6486  ‘cfv 6492  (class class class)co 7358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7315  df-ov 7361  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-oi 9415

This theorem is referenced by:  fpwwe  10557  canthwelem  10561  pwfseqlem4  10573