Theorem fpwwe2 10572

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 9959. 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 10569	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊:dom 𝑊⟶𝒫 (𝑋 × 𝑋))
6	5	ffund 6674	. . . . . . . . 9 ⊢ (𝜑 → Fun 𝑊)
7		funbrfv2b 6900	. . . . . . . . 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 4897	. . . . . . 7 ⊢ (𝑌 ∈ dom 𝑊 → 𝑌 ⊆ ∪ dom 𝑊)
12	11, 4	sseqtrrdi 3985	. . . . . 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 10570	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑋 ∈ dom 𝑊)
20		funfvbrb 7005	. . . . . . . . . . . . . . . . . . . 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 10561	. . . . . . . . . . . . . . . . . 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 5274	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → 𝑋 ∈ V)
29	28	difexd 5281	. . . . . . . . . . . 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 5621	. . . . . . . . . . . . 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 4096	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (𝑋 ∖ 𝑌) ⊆ 𝑋)
35		fri 5589	. . . . . . . . . . . . 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 4325	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∩ (◡(𝑊‘𝑋) “ {𝑧})) ⊆ 𝑌 ↔ ((𝑋 ∩ (◡(𝑊‘𝑋) “ {𝑧})) ∖ 𝑌) = ∅)
39		indif1 4241	. . . . . . . . . . . . . . . 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 4409	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∖ 𝑌) ∩ (◡(𝑊‘𝑋) “ {𝑧})) = ∅ ↔ ∀𝑤 ∈ (𝑋 ∖ 𝑌) ¬ 𝑤 ∈ (◡(𝑊‘𝑋) “ {𝑧}))
42		vex 3448	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑤 ∈ V
43	42	eliniseg 6054	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ V → (𝑤 ∈ (◡(𝑊‘𝑋) “ {𝑧}) ↔ 𝑤(𝑊‘𝑋)𝑧))
44	43	elv 3449	. . . . . . . . . . . . . . . . . 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 6042	. . . . . . . . . . . . . . . . 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 5856	. . . . . . . . . . . . . . . . . . 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 5883	. . . . . . . . . . . . . . . . . 18 ⊢ dom (𝑋 × 𝑋) = 𝑋
54	52, 53	sseqtrdi 3984	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → dom (𝑊‘𝑋) ⊆ 𝑋)
55	49, 54	sstrid 3955	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑋)
56		sseqin2 4182	. . . . . . . . . . . . . . . 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 3975	. . . . . . . . . . . . . 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 4091	. . . . . . . . . . . . . . . . . . . . . . . . 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 5113	. . . . . . . . . . . . . . . . . . . . . . . 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 4090	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5680	. . . . . . . . . . . . . . . . . . . . . . . . . 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 5154	. . . . . . . . . . . . . . . . . . . . . . . . . 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 10561	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5145	. . . . . . . . . . . . . . . . . . . . . . . 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 5680	. . . . . . . . . . . . . . . . . . . . . . . . 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 5622	. . . . . . . . . . . . . . . . . . . . . . 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 3943	. . . . . . . . . . . . . . . . . . . . . 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 5569	. . . . . . . . . . . . . . . . . . . . . . 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 3949	. . . . . . . . . . . . . . . . 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 3961	. . . . . . . . . . . . . . . 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 4189	. . . . . . . . . . . . . . . . . 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 4178	. . . . . . . . . . . . . . . . . . 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 3929	. . . . . . . . . . . . . . . . . . . 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 4197	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑊‘𝑋) ∩ (𝑌 × 𝑌)) ⊆ (𝑌 × 𝑌)
117		xpss1 5650	. . . . . . . . . . . . . . . . . . . . 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 3955	. . . . . . . . . . . . . . . . . . 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 3929	. . . . . . . . . . . . . . . . . . 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 5663	. . . . . . . . . . . . . . . . . 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 4179	. . . . . . . . . . . . . . . . 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 7387	. . . . . . . . . . . . . . 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 10562	. . . . . . . . . . . . . . . 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 4325	. . . . . . . 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 10568	. . . . . 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 3961	. . . 4 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑌 = 𝑋)
147	6	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → Fun 𝑊)
148	146, 143	eqbrtrrd 5126	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑋𝑊𝑅)
149		funbrfv 6891	. . . . . 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 10571	. . . 4 ⊢ (𝜑 → (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋)
155	22, 154	jca 511	. . 3 ⊢ (𝜑 → (𝑋𝑊(𝑊‘𝑋) ∧ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋))
156		breq12 5107	. . . 4 ⊢ ((𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋)) → (𝑌𝑊𝑅 ↔ 𝑋𝑊(𝑊‘𝑋)))
157		oveq12 7378	. . . . 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 3444  [wsbc 3750   ∖ cdif 3908   ∩ cin 3910   ⊆ wss 3911  ∅c0 4292  𝒫 cpw 4559  {csn 4585  ∪ cuni 4867   class class class wbr 5102  {copab 5164   Or wor 5538   Fr wfr 5581   We wwe 5583   × cxp 5629  ^◡ccnv 5630  dom cdm 5631   “ cima 5634  Fun wfun 6493  ‘cfv 6499  (class class class)co 7369

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 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

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 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7326  df-ov 7372  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-oi 9439

This theorem is referenced by:  fpwwe  10575  canthwelem  10579  pwfseqlem4  10591