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Theorem fpwwe2 10684
Description: Given any function 𝐹 from well-orderings of subsets of 𝐴 to 𝐴, there is a unique well-ordered subset 𝑋, (𝑊𝑋)⟩ 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 10071. 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⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
fpwwe2.2 (𝜑𝐴𝑉)
fpwwe2.3 ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
fpwwe2.4 𝑋 = dom 𝑊
Assertion
Ref Expression
fpwwe2 (𝜑 → ((𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌) ↔ (𝑌 = 𝑋𝑅 = (𝑊𝑋))))
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.
StepHypRef Expression
1 fpwwe2.1 . . . . . . . . . . 11 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
2 fpwwe2.2 . . . . . . . . . . 11 (𝜑𝐴𝑉)
3 fpwwe2.3 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
4 fpwwe2.4 . . . . . . . . . . 11 𝑋 = dom 𝑊
51, 2, 3, 4fpwwe2lem10 10681 . . . . . . . . . 10 (𝜑𝑊:dom 𝑊⟶𝒫 (𝑋 × 𝑋))
65ffund 6739 . . . . . . . . 9 (𝜑 → Fun 𝑊)
7 funbrfv2b 6965 . . . . . . . . 9 (Fun 𝑊 → (𝑌𝑊𝑅 ↔ (𝑌 ∈ dom 𝑊 ∧ (𝑊𝑌) = 𝑅)))
86, 7syl 17 . . . . . . . 8 (𝜑 → (𝑌𝑊𝑅 ↔ (𝑌 ∈ dom 𝑊 ∧ (𝑊𝑌) = 𝑅)))
98simprbda 498 . . . . . . 7 ((𝜑𝑌𝑊𝑅) → 𝑌 ∈ dom 𝑊)
109adantrr 717 . . . . . 6 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑌 ∈ dom 𝑊)
11 elssuni 4936 . . . . . . 7 (𝑌 ∈ dom 𝑊𝑌 dom 𝑊)
1211, 4sseqtrrdi 4024 . . . . . 6 (𝑌 ∈ dom 𝑊𝑌𝑋)
1310, 12syl 17 . . . . 5 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑌𝑋)
14 simpl 482 . . . . . . 7 ((𝑋𝑌 ∧ (𝑊𝑋) = (𝑅 ∩ (𝑌 × 𝑋))) → 𝑋𝑌)
1514a1i 11 . . . . . 6 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → ((𝑋𝑌 ∧ (𝑊𝑋) = (𝑅 ∩ (𝑌 × 𝑋))) → 𝑋𝑌))
16 simplrr 777 . . . . . . . . 9 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑌𝐹𝑅) ∈ 𝑌)
172adantr 480 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝐴𝑉)
1817adantr 480 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → 𝐴𝑉)
191, 2, 3, 4fpwwe2lem11 10682 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑋 ∈ dom 𝑊)
20 funfvbrb 7070 . . . . . . . . . . . . . . . . . . . 20 (Fun 𝑊 → (𝑋 ∈ dom 𝑊𝑋𝑊(𝑊𝑋)))
216, 20syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑋 ∈ dom 𝑊𝑋𝑊(𝑊𝑋)))
2219, 21mpbid 232 . . . . . . . . . . . . . . . . . 18 (𝜑𝑋𝑊(𝑊𝑋))
231, 2fpwwe2lem2 10673 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑋𝑊(𝑊𝑋) ↔ ((𝑋𝐴 ∧ (𝑊𝑋) ⊆ (𝑋 × 𝑋)) ∧ ((𝑊𝑋) We 𝑋 ∧ ∀𝑦𝑋 [((𝑊𝑋) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊𝑋) ∩ (𝑢 × 𝑢))) = 𝑦))))
2422, 23mpbid 232 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑋𝐴 ∧ (𝑊𝑋) ⊆ (𝑋 × 𝑋)) ∧ ((𝑊𝑋) We 𝑋 ∧ ∀𝑦𝑋 [((𝑊𝑋) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊𝑋) ∩ (𝑢 × 𝑢))) = 𝑦)))
2524ad2antrr 726 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑋𝐴 ∧ (𝑊𝑋) ⊆ (𝑋 × 𝑋)) ∧ ((𝑊𝑋) We 𝑋 ∧ ∀𝑦𝑋 [((𝑊𝑋) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊𝑋) ∩ (𝑢 × 𝑢))) = 𝑦)))
2625simpld 494 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑋𝐴 ∧ (𝑊𝑋) ⊆ (𝑋 × 𝑋)))
2726simpld 494 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → 𝑋𝐴)
2818, 27ssexd 5323 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → 𝑋 ∈ V)
2928difexd 5330 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑋𝑌) ∈ V)
3025simprd 495 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑊𝑋) We 𝑋 ∧ ∀𝑦𝑋 [((𝑊𝑋) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊𝑋) ∩ (𝑢 × 𝑢))) = 𝑦))
3130simpld 494 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑊𝑋) We 𝑋)
32 wefr 5674 . . . . . . . . . . . . 13 ((𝑊𝑋) We 𝑋 → (𝑊𝑋) Fr 𝑋)
3331, 32syl 17 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑊𝑋) Fr 𝑋)
34 difssd 4136 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑋𝑌) ⊆ 𝑋)
35 fri 5641 . . . . . . . . . . . . 13 ((((𝑋𝑌) ∈ V ∧ (𝑊𝑋) Fr 𝑋) ∧ ((𝑋𝑌) ⊆ 𝑋 ∧ (𝑋𝑌) ≠ ∅)) → ∃𝑧 ∈ (𝑋𝑌)∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤(𝑊𝑋)𝑧)
3635expr 456 . . . . . . . . . . . 12 ((((𝑋𝑌) ∈ V ∧ (𝑊𝑋) Fr 𝑋) ∧ (𝑋𝑌) ⊆ 𝑋) → ((𝑋𝑌) ≠ ∅ → ∃𝑧 ∈ (𝑋𝑌)∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤(𝑊𝑋)𝑧))
3729, 33, 34, 36syl21anc 837 . . . . . . . . . . 11 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑋𝑌) ≠ ∅ → ∃𝑧 ∈ (𝑋𝑌)∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤(𝑊𝑋)𝑧))
38 ssdif0 4365 . . . . . . . . . . . . . . 15 ((𝑋 ∩ ((𝑊𝑋) “ {𝑧})) ⊆ 𝑌 ↔ ((𝑋 ∩ ((𝑊𝑋) “ {𝑧})) ∖ 𝑌) = ∅)
39 indif1 4281 . . . . . . . . . . . . . . . 16 ((𝑋𝑌) ∩ ((𝑊𝑋) “ {𝑧})) = ((𝑋 ∩ ((𝑊𝑋) “ {𝑧})) ∖ 𝑌)
4039eqeq1i 2741 . . . . . . . . . . . . . . 15 (((𝑋𝑌) ∩ ((𝑊𝑋) “ {𝑧})) = ∅ ↔ ((𝑋 ∩ ((𝑊𝑋) “ {𝑧})) ∖ 𝑌) = ∅)
41 disj 4449 . . . . . . . . . . . . . . . 16 (((𝑋𝑌) ∩ ((𝑊𝑋) “ {𝑧})) = ∅ ↔ ∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤 ∈ ((𝑊𝑋) “ {𝑧}))
42 vex 3483 . . . . . . . . . . . . . . . . . . . 20 𝑤 ∈ V
4342eliniseg 6111 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ V → (𝑤 ∈ ((𝑊𝑋) “ {𝑧}) ↔ 𝑤(𝑊𝑋)𝑧))
4443elv 3484 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ ((𝑊𝑋) “ {𝑧}) ↔ 𝑤(𝑊𝑋)𝑧)
4544notbii 320 . . . . . . . . . . . . . . . . 17 𝑤 ∈ ((𝑊𝑋) “ {𝑧}) ↔ ¬ 𝑤(𝑊𝑋)𝑧)
4645ralbii 3092 . . . . . . . . . . . . . . . 16 (∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤 ∈ ((𝑊𝑋) “ {𝑧}) ↔ ∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤(𝑊𝑋)𝑧)
4741, 46bitri 275 . . . . . . . . . . . . . . 15 (((𝑋𝑌) ∩ ((𝑊𝑋) “ {𝑧})) = ∅ ↔ ∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤(𝑊𝑋)𝑧)
4838, 40, 473bitr2i 299 . . . . . . . . . . . . . 14 ((𝑋 ∩ ((𝑊𝑋) “ {𝑧})) ⊆ 𝑌 ↔ ∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤(𝑊𝑋)𝑧)
49 cnvimass 6099 . . . . . . . . . . . . . . . . 17 ((𝑊𝑋) “ {𝑧}) ⊆ dom (𝑊𝑋)
5026simprd 495 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑊𝑋) ⊆ (𝑋 × 𝑋))
51 dmss 5912 . . . . . . . . . . . . . . . . . . 19 ((𝑊𝑋) ⊆ (𝑋 × 𝑋) → dom (𝑊𝑋) ⊆ dom (𝑋 × 𝑋))
5250, 51syl 17 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → dom (𝑊𝑋) ⊆ dom (𝑋 × 𝑋))
53 dmxpid 5940 . . . . . . . . . . . . . . . . . 18 dom (𝑋 × 𝑋) = 𝑋
5452, 53sseqtrdi 4023 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → dom (𝑊𝑋) ⊆ 𝑋)
5549, 54sstrid 3994 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑊𝑋) “ {𝑧}) ⊆ 𝑋)
56 sseqin2 4222 . . . . . . . . . . . . . . . 16 (((𝑊𝑋) “ {𝑧}) ⊆ 𝑋 ↔ (𝑋 ∩ ((𝑊𝑋) “ {𝑧})) = ((𝑊𝑋) “ {𝑧}))
5755, 56sylib 218 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑋 ∩ ((𝑊𝑋) “ {𝑧})) = ((𝑊𝑋) “ {𝑧}))
5857sseq1d 4014 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑋 ∩ ((𝑊𝑋) “ {𝑧})) ⊆ 𝑌 ↔ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌))
5948, 58bitr3id 285 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤(𝑊𝑋)𝑧 ↔ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌))
6059rexbidv 3178 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (∃𝑧 ∈ (𝑋𝑌)∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤(𝑊𝑋)𝑧 ↔ ∃𝑧 ∈ (𝑋𝑌)((𝑊𝑋) “ {𝑧}) ⊆ 𝑌))
61 eldifn 4131 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 ∈ (𝑋𝑌) → ¬ 𝑧𝑌)
6261ad2antrl 728 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → ¬ 𝑧𝑌)
63 eleq1w 2823 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = 𝑧 → (𝑤𝑌𝑧𝑌))
6463notbid 318 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = 𝑧 → (¬ 𝑤𝑌 ↔ ¬ 𝑧𝑌))
6562, 64syl5ibrcom 247 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑤 = 𝑧 → ¬ 𝑤𝑌))
6665con2d 134 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑤𝑌 → ¬ 𝑤 = 𝑧))
6766imp 406 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → ¬ 𝑤 = 𝑧)
6862adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → ¬ 𝑧𝑌)
69 simprr 772 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → 𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))
7069ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → 𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))
7170breqd 5153 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → (𝑧𝑅𝑤𝑧((𝑊𝑋) ∩ (𝑋 × 𝑌))𝑤))
72 eldifi 4130 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 ∈ (𝑋𝑌) → 𝑧𝑋)
7372ad2antrl 728 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑧𝑋)
7473adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → 𝑧𝑋)
75 simpr 484 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → 𝑤𝑌)
76 brxp 5733 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧(𝑋 × 𝑌)𝑤 ↔ (𝑧𝑋𝑤𝑌))
7774, 75, 76sylanbrc 583 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → 𝑧(𝑋 × 𝑌)𝑤)
78 brin 5194 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧((𝑊𝑋) ∩ (𝑋 × 𝑌))𝑤 ↔ (𝑧(𝑊𝑋)𝑤𝑧(𝑋 × 𝑌)𝑤))
7978rbaib 538 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧(𝑋 × 𝑌)𝑤 → (𝑧((𝑊𝑋) ∩ (𝑋 × 𝑌))𝑤𝑧(𝑊𝑋)𝑤))
8077, 79syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → (𝑧((𝑊𝑋) ∩ (𝑋 × 𝑌))𝑤𝑧(𝑊𝑋)𝑤))
8171, 80bitrd 279 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → (𝑧𝑅𝑤𝑧(𝑊𝑋)𝑤))
821, 2fpwwe2lem2 10673 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑 → (𝑌𝑊𝑅 ↔ ((𝑌𝐴𝑅 ⊆ (𝑌 × 𝑌)) ∧ (𝑅 We 𝑌 ∧ ∀𝑦𝑌 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))))
8382biimpa 476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑌𝑊𝑅) → ((𝑌𝐴𝑅 ⊆ (𝑌 × 𝑌)) ∧ (𝑅 We 𝑌 ∧ ∀𝑦𝑌 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦)))
8483adantrr 717 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → ((𝑌𝐴𝑅 ⊆ (𝑌 × 𝑌)) ∧ (𝑅 We 𝑌 ∧ ∀𝑦𝑌 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦)))
8584simpld 494 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → (𝑌𝐴𝑅 ⊆ (𝑌 × 𝑌)))
8685simprd 495 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑅 ⊆ (𝑌 × 𝑌))
8786ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → 𝑅 ⊆ (𝑌 × 𝑌))
8887ssbrd 5185 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → (𝑧𝑅𝑤𝑧(𝑌 × 𝑌)𝑤))
89 brxp 5733 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧(𝑌 × 𝑌)𝑤 ↔ (𝑧𝑌𝑤𝑌))
9089simplbi 497 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧(𝑌 × 𝑌)𝑤𝑧𝑌)
9188, 90syl6 35 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → (𝑧𝑅𝑤𝑧𝑌))
9281, 91sylbird 260 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → (𝑧(𝑊𝑋)𝑤𝑧𝑌))
9368, 92mtod 198 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → ¬ 𝑧(𝑊𝑋)𝑤)
9431ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → (𝑊𝑋) We 𝑋)
95 weso 5675 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑊𝑋) We 𝑋 → (𝑊𝑋) Or 𝑋)
9694, 95syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → (𝑊𝑋) Or 𝑋)
9713ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑌𝑋)
9897sselda 3982 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → 𝑤𝑋)
99 sotric 5621 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑊𝑋) Or 𝑋 ∧ (𝑤𝑋𝑧𝑋)) → (𝑤(𝑊𝑋)𝑧 ↔ ¬ (𝑤 = 𝑧𝑧(𝑊𝑋)𝑤)))
100 ioran 985 . . . . . . . . . . . . . . . . . . . . . . 23 (¬ (𝑤 = 𝑧𝑧(𝑊𝑋)𝑤) ↔ (¬ 𝑤 = 𝑧 ∧ ¬ 𝑧(𝑊𝑋)𝑤))
10199, 100bitrdi 287 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑊𝑋) Or 𝑋 ∧ (𝑤𝑋𝑧𝑋)) → (𝑤(𝑊𝑋)𝑧 ↔ (¬ 𝑤 = 𝑧 ∧ ¬ 𝑧(𝑊𝑋)𝑤)))
10296, 98, 74, 101syl12anc 836 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → (𝑤(𝑊𝑋)𝑧 ↔ (¬ 𝑤 = 𝑧 ∧ ¬ 𝑧(𝑊𝑋)𝑤)))
10367, 93, 102mpbir2and 713 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → 𝑤(𝑊𝑋)𝑧)
104103, 44sylibr 234 . . . . . . . . . . . . . . . . . . 19 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → 𝑤 ∈ ((𝑊𝑋) “ {𝑧}))
105104ex 412 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑤𝑌𝑤 ∈ ((𝑊𝑋) “ {𝑧})))
106105ssrdv 3988 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑌 ⊆ ((𝑊𝑋) “ {𝑧}))
107 simprr 772 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)
108106, 107eqssd 4000 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑌 = ((𝑊𝑋) “ {𝑧}))
109 in32 4229 . . . . . . . . . . . . . . . . . 18 (((𝑊𝑋) ∩ (𝑋 × 𝑌)) ∩ (𝑌 × 𝑌)) = (((𝑊𝑋) ∩ (𝑌 × 𝑌)) ∩ (𝑋 × 𝑌))
110 simplrr 777 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))
111110ineq1d 4218 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑅 ∩ (𝑌 × 𝑌)) = (((𝑊𝑋) ∩ (𝑋 × 𝑌)) ∩ (𝑌 × 𝑌)))
11286ad2antrr 726 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑅 ⊆ (𝑌 × 𝑌))
113 dfss2 3968 . . . . . . . . . . . . . . . . . . . 20 (𝑅 ⊆ (𝑌 × 𝑌) ↔ (𝑅 ∩ (𝑌 × 𝑌)) = 𝑅)
114112, 113sylib 218 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑅 ∩ (𝑌 × 𝑌)) = 𝑅)
115111, 114eqtr3d 2778 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (((𝑊𝑋) ∩ (𝑋 × 𝑌)) ∩ (𝑌 × 𝑌)) = 𝑅)
116 inss2 4237 . . . . . . . . . . . . . . . . . . . 20 ((𝑊𝑋) ∩ (𝑌 × 𝑌)) ⊆ (𝑌 × 𝑌)
117 xpss1 5703 . . . . . . . . . . . . . . . . . . . . 21 (𝑌𝑋 → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑌))
11897, 117syl 17 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑌))
119116, 118sstrid 3994 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → ((𝑊𝑋) ∩ (𝑌 × 𝑌)) ⊆ (𝑋 × 𝑌))
120 dfss2 3968 . . . . . . . . . . . . . . . . . . 19 (((𝑊𝑋) ∩ (𝑌 × 𝑌)) ⊆ (𝑋 × 𝑌) ↔ (((𝑊𝑋) ∩ (𝑌 × 𝑌)) ∩ (𝑋 × 𝑌)) = ((𝑊𝑋) ∩ (𝑌 × 𝑌)))
121119, 120sylib 218 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (((𝑊𝑋) ∩ (𝑌 × 𝑌)) ∩ (𝑋 × 𝑌)) = ((𝑊𝑋) ∩ (𝑌 × 𝑌)))
122109, 115, 1213eqtr3a 2800 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑅 = ((𝑊𝑋) ∩ (𝑌 × 𝑌)))
123108sqxpeqd 5716 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑌 × 𝑌) = (((𝑊𝑋) “ {𝑧}) × ((𝑊𝑋) “ {𝑧})))
124123ineq2d 4219 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → ((𝑊𝑋) ∩ (𝑌 × 𝑌)) = ((𝑊𝑋) ∩ (((𝑊𝑋) “ {𝑧}) × ((𝑊𝑋) “ {𝑧}))))
125122, 124eqtrd 2776 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑅 = ((𝑊𝑋) ∩ (((𝑊𝑋) “ {𝑧}) × ((𝑊𝑋) “ {𝑧}))))
126108, 125oveq12d 7450 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑌𝐹𝑅) = (((𝑊𝑋) “ {𝑧})𝐹((𝑊𝑋) ∩ (((𝑊𝑋) “ {𝑧}) × ((𝑊𝑋) “ {𝑧})))))
12718adantr 480 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝐴𝑉)
12822adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑋𝑊(𝑊𝑋))
129128ad2antrr 726 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑋𝑊(𝑊𝑋))
1301, 127, 129fpwwe2lem3 10674 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑧𝑋) → (((𝑊𝑋) “ {𝑧})𝐹((𝑊𝑋) ∩ (((𝑊𝑋) “ {𝑧}) × ((𝑊𝑋) “ {𝑧})))) = 𝑧)
13173, 130mpdan 687 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (((𝑊𝑋) “ {𝑧})𝐹((𝑊𝑋) ∩ (((𝑊𝑋) “ {𝑧}) × ((𝑊𝑋) “ {𝑧})))) = 𝑧)
132126, 131eqtrd 2776 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑌𝐹𝑅) = 𝑧)
133132, 62eqneltrd 2860 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → ¬ (𝑌𝐹𝑅) ∈ 𝑌)
134133rexlimdvaa 3155 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (∃𝑧 ∈ (𝑋𝑌)((𝑊𝑋) “ {𝑧}) ⊆ 𝑌 → ¬ (𝑌𝐹𝑅) ∈ 𝑌))
13560, 134sylbid 240 . . . . . . . . . . 11 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (∃𝑧 ∈ (𝑋𝑌)∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤(𝑊𝑋)𝑧 → ¬ (𝑌𝐹𝑅) ∈ 𝑌))
13637, 135syld 47 . . . . . . . . . 10 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑋𝑌) ≠ ∅ → ¬ (𝑌𝐹𝑅) ∈ 𝑌))
137136necon4ad 2958 . . . . . . . . 9 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑌𝐹𝑅) ∈ 𝑌 → (𝑋𝑌) = ∅))
13816, 137mpd 15 . . . . . . . 8 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑋𝑌) = ∅)
139 ssdif0 4365 . . . . . . . 8 (𝑋𝑌 ↔ (𝑋𝑌) = ∅)
140138, 139sylibr 234 . . . . . . 7 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → 𝑋𝑌)
141140ex 412 . . . . . 6 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → ((𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌))) → 𝑋𝑌))
1423adantlr 715 . . . . . . 7 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
143 simprl 770 . . . . . . 7 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑌𝑊𝑅)
1441, 17, 142, 128, 143fpwwe2lem9 10680 . . . . . 6 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → ((𝑋𝑌 ∧ (𝑊𝑋) = (𝑅 ∩ (𝑌 × 𝑋))) ∨ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))))
14515, 141, 144mpjaod 860 . . . . 5 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑋𝑌)
14613, 145eqssd 4000 . . . 4 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑌 = 𝑋)
1476adantr 480 . . . . . 6 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → Fun 𝑊)
148146, 143eqbrtrrd 5166 . . . . . 6 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑋𝑊𝑅)
149 funbrfv 6956 . . . . . 6 (Fun 𝑊 → (𝑋𝑊𝑅 → (𝑊𝑋) = 𝑅))
150147, 148, 149sylc 65 . . . . 5 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → (𝑊𝑋) = 𝑅)
151150eqcomd 2742 . . . 4 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑅 = (𝑊𝑋))
152146, 151jca 511 . . 3 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → (𝑌 = 𝑋𝑅 = (𝑊𝑋)))
153152ex 412 . 2 (𝜑 → ((𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌) → (𝑌 = 𝑋𝑅 = (𝑊𝑋))))
1541, 2, 3, 4fpwwe2lem12 10683 . . . 4 (𝜑 → (𝑋𝐹(𝑊𝑋)) ∈ 𝑋)
15522, 154jca 511 . . 3 (𝜑 → (𝑋𝑊(𝑊𝑋) ∧ (𝑋𝐹(𝑊𝑋)) ∈ 𝑋))
156 breq12 5147 . . . 4 ((𝑌 = 𝑋𝑅 = (𝑊𝑋)) → (𝑌𝑊𝑅𝑋𝑊(𝑊𝑋)))
157 oveq12 7441 . . . . 5 ((𝑌 = 𝑋𝑅 = (𝑊𝑋)) → (𝑌𝐹𝑅) = (𝑋𝐹(𝑊𝑋)))
158 simpl 482 . . . . 5 ((𝑌 = 𝑋𝑅 = (𝑊𝑋)) → 𝑌 = 𝑋)
159157, 158eleq12d 2834 . . . 4 ((𝑌 = 𝑋𝑅 = (𝑊𝑋)) → ((𝑌𝐹𝑅) ∈ 𝑌 ↔ (𝑋𝐹(𝑊𝑋)) ∈ 𝑋))
160156, 159anbi12d 632 . . 3 ((𝑌 = 𝑋𝑅 = (𝑊𝑋)) → ((𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌) ↔ (𝑋𝑊(𝑊𝑋) ∧ (𝑋𝐹(𝑊𝑋)) ∈ 𝑋)))
161155, 160syl5ibrcom 247 . 2 (𝜑 → ((𝑌 = 𝑋𝑅 = (𝑊𝑋)) → (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)))
162153, 161impbid 212 1 (𝜑 → ((𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌) ↔ (𝑌 = 𝑋𝑅 = (𝑊𝑋))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1539  wcel 2107  wne 2939  wral 3060  wrex 3069  Vcvv 3479  [wsbc 3787  cdif 3947  cin 3949  wss 3950  c0 4332  𝒫 cpw 4599  {csn 4625   cuni 4906   class class class wbr 5142  {copab 5204   Or wor 5590   Fr wfr 5633   We wwe 5635   × cxp 5682  ccnv 5683  dom cdm 5684  cima 5687  Fun wfun 6554  cfv 6560  (class class class)co 7432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-oi 9551
This theorem is referenced by:  fpwwe  10687  canthwelem  10691  pwfseqlem4  10703
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