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Theorem fpwwe2 10686
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 10073. 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 10683 . . . . . . . . . 10 (𝜑𝑊:dom 𝑊⟶𝒫 (𝑋 × 𝑋))
65ffund 6732 . . . . . . . . 9 (𝜑 → Fun 𝑊)
7 funbrfv2b 6960 . . . . . . . . 9 (Fun 𝑊 → (𝑌𝑊𝑅 ↔ (𝑌 ∈ dom 𝑊 ∧ (𝑊𝑌) = 𝑅)))
86, 7syl 17 . . . . . . . 8 (𝜑 → (𝑌𝑊𝑅 ↔ (𝑌 ∈ dom 𝑊 ∧ (𝑊𝑌) = 𝑅)))
98simprbda 497 . . . . . . 7 ((𝜑𝑌𝑊𝑅) → 𝑌 ∈ dom 𝑊)
109adantrr 715 . . . . . 6 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑌 ∈ dom 𝑊)
11 elssuni 4945 . . . . . . 7 (𝑌 ∈ dom 𝑊𝑌 dom 𝑊)
1211, 4sseqtrrdi 4031 . . . . . 6 (𝑌 ∈ dom 𝑊𝑌𝑋)
1310, 12syl 17 . . . . 5 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑌𝑋)
14 simpl 481 . . . . . . 7 ((𝑋𝑌 ∧ (𝑊𝑋) = (𝑅 ∩ (𝑌 × 𝑋))) → 𝑋𝑌)
1514a1i 11 . . . . . 6 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → ((𝑋𝑌 ∧ (𝑊𝑋) = (𝑅 ∩ (𝑌 × 𝑋))) → 𝑋𝑌))
16 simplrr 776 . . . . . . . . 9 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑌𝐹𝑅) ∈ 𝑌)
172adantr 479 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝐴𝑉)
1817adantr 479 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → 𝐴𝑉)
191, 2, 3, 4fpwwe2lem11 10684 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑋 ∈ dom 𝑊)
20 funfvbrb 7064 . . . . . . . . . . . . . . . . . . . 20 (Fun 𝑊 → (𝑋 ∈ dom 𝑊𝑋𝑊(𝑊𝑋)))
216, 20syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑋 ∈ dom 𝑊𝑋𝑊(𝑊𝑋)))
2219, 21mpbid 231 . . . . . . . . . . . . . . . . . 18 (𝜑𝑋𝑊(𝑊𝑋))
231, 2fpwwe2lem2 10675 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑋𝑊(𝑊𝑋) ↔ ((𝑋𝐴 ∧ (𝑊𝑋) ⊆ (𝑋 × 𝑋)) ∧ ((𝑊𝑋) We 𝑋 ∧ ∀𝑦𝑋 [((𝑊𝑋) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊𝑋) ∩ (𝑢 × 𝑢))) = 𝑦))))
2422, 23mpbid 231 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑋𝐴 ∧ (𝑊𝑋) ⊆ (𝑋 × 𝑋)) ∧ ((𝑊𝑋) We 𝑋 ∧ ∀𝑦𝑋 [((𝑊𝑋) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊𝑋) ∩ (𝑢 × 𝑢))) = 𝑦)))
2524ad2antrr 724 . . . . . . . . . . . . . . . 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 493 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑋𝐴 ∧ (𝑊𝑋) ⊆ (𝑋 × 𝑋)))
2726simpld 493 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → 𝑋𝐴)
2818, 27ssexd 5329 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → 𝑋 ∈ V)
2928difexd 5336 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑋𝑌) ∈ V)
3025simprd 494 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑊𝑋) We 𝑋 ∧ ∀𝑦𝑋 [((𝑊𝑋) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊𝑋) ∩ (𝑢 × 𝑢))) = 𝑦))
3130simpld 493 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑊𝑋) We 𝑋)
32 wefr 5672 . . . . . . . . . . . . 13 ((𝑊𝑋) We 𝑋 → (𝑊𝑋) Fr 𝑋)
3331, 32syl 17 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑊𝑋) Fr 𝑋)
34 difssd 4132 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑋𝑌) ⊆ 𝑋)
35 fri 5642 . . . . . . . . . . . . 13 ((((𝑋𝑌) ∈ V ∧ (𝑊𝑋) Fr 𝑋) ∧ ((𝑋𝑌) ⊆ 𝑋 ∧ (𝑋𝑌) ≠ ∅)) → ∃𝑧 ∈ (𝑋𝑌)∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤(𝑊𝑋)𝑧)
3635expr 455 . . . . . . . . . . . 12 ((((𝑋𝑌) ∈ V ∧ (𝑊𝑋) Fr 𝑋) ∧ (𝑋𝑌) ⊆ 𝑋) → ((𝑋𝑌) ≠ ∅ → ∃𝑧 ∈ (𝑋𝑌)∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤(𝑊𝑋)𝑧))
3729, 33, 34, 36syl21anc 836 . . . . . . . . . . 11 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑋𝑌) ≠ ∅ → ∃𝑧 ∈ (𝑋𝑌)∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤(𝑊𝑋)𝑧))
38 ssdif0 4366 . . . . . . . . . . . . . . 15 ((𝑋 ∩ ((𝑊𝑋) “ {𝑧})) ⊆ 𝑌 ↔ ((𝑋 ∩ ((𝑊𝑋) “ {𝑧})) ∖ 𝑌) = ∅)
39 indif1 4273 . . . . . . . . . . . . . . . 16 ((𝑋𝑌) ∩ ((𝑊𝑋) “ {𝑧})) = ((𝑋 ∩ ((𝑊𝑋) “ {𝑧})) ∖ 𝑌)
4039eqeq1i 2731 . . . . . . . . . . . . . . 15 (((𝑋𝑌) ∩ ((𝑊𝑋) “ {𝑧})) = ∅ ↔ ((𝑋 ∩ ((𝑊𝑋) “ {𝑧})) ∖ 𝑌) = ∅)
41 disj 4452 . . . . . . . . . . . . . . . 16 (((𝑋𝑌) ∩ ((𝑊𝑋) “ {𝑧})) = ∅ ↔ ∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤 ∈ ((𝑊𝑋) “ {𝑧}))
42 vex 3466 . . . . . . . . . . . . . . . . . . . 20 𝑤 ∈ V
4342eliniseg 6104 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ V → (𝑤 ∈ ((𝑊𝑋) “ {𝑧}) ↔ 𝑤(𝑊𝑋)𝑧))
4443elv 3468 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ ((𝑊𝑋) “ {𝑧}) ↔ 𝑤(𝑊𝑋)𝑧)
4544notbii 319 . . . . . . . . . . . . . . . . 17 𝑤 ∈ ((𝑊𝑋) “ {𝑧}) ↔ ¬ 𝑤(𝑊𝑋)𝑧)
4645ralbii 3083 . . . . . . . . . . . . . . . 16 (∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤 ∈ ((𝑊𝑋) “ {𝑧}) ↔ ∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤(𝑊𝑋)𝑧)
4741, 46bitri 274 . . . . . . . . . . . . . . 15 (((𝑋𝑌) ∩ ((𝑊𝑋) “ {𝑧})) = ∅ ↔ ∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤(𝑊𝑋)𝑧)
4838, 40, 473bitr2i 298 . . . . . . . . . . . . . 14 ((𝑋 ∩ ((𝑊𝑋) “ {𝑧})) ⊆ 𝑌 ↔ ∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤(𝑊𝑋)𝑧)
49 cnvimass 6091 . . . . . . . . . . . . . . . . 17 ((𝑊𝑋) “ {𝑧}) ⊆ dom (𝑊𝑋)
5026simprd 494 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑊𝑋) ⊆ (𝑋 × 𝑋))
51 dmss 5909 . . . . . . . . . . . . . . . . . . 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 5936 . . . . . . . . . . . . . . . . . 18 dom (𝑋 × 𝑋) = 𝑋
5452, 53sseqtrdi 4030 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → dom (𝑊𝑋) ⊆ 𝑋)
5549, 54sstrid 3991 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑊𝑋) “ {𝑧}) ⊆ 𝑋)
56 sseqin2 4216 . . . . . . . . . . . . . . . 16 (((𝑊𝑋) “ {𝑧}) ⊆ 𝑋 ↔ (𝑋 ∩ ((𝑊𝑋) “ {𝑧})) = ((𝑊𝑋) “ {𝑧}))
5755, 56sylib 217 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (𝑋 ∩ ((𝑊𝑋) “ {𝑧})) = ((𝑊𝑋) “ {𝑧}))
5857sseq1d 4011 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑋 ∩ ((𝑊𝑋) “ {𝑧})) ⊆ 𝑌 ↔ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌))
5948, 58bitr3id 284 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤(𝑊𝑋)𝑧 ↔ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌))
6059rexbidv 3169 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (∃𝑧 ∈ (𝑋𝑌)∀𝑤 ∈ (𝑋𝑌) ¬ 𝑤(𝑊𝑋)𝑧 ↔ ∃𝑧 ∈ (𝑋𝑌)((𝑊𝑋) “ {𝑧}) ⊆ 𝑌))
61 eldifn 4127 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 ∈ (𝑋𝑌) → ¬ 𝑧𝑌)
6261ad2antrl 726 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → ¬ 𝑧𝑌)
63 eleq1w 2809 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = 𝑧 → (𝑤𝑌𝑧𝑌))
6463notbid 317 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = 𝑧 → (¬ 𝑤𝑌 ↔ ¬ 𝑧𝑌))
6562, 64syl5ibrcom 246 . . . . . . . . . . . . . . . . . . . . . . 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 405 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → ¬ 𝑤 = 𝑧)
6862adantr 479 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → ¬ 𝑧𝑌)
69 simprr 771 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → 𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))
7069ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → 𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))
7170breqd 5164 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → (𝑧𝑅𝑤𝑧((𝑊𝑋) ∩ (𝑋 × 𝑌))𝑤))
72 eldifi 4126 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 ∈ (𝑋𝑌) → 𝑧𝑋)
7372ad2antrl 726 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑧𝑋)
7473adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → 𝑧𝑋)
75 simpr 483 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → 𝑤𝑌)
76 brxp 5731 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧(𝑋 × 𝑌)𝑤 ↔ (𝑧𝑋𝑤𝑌))
7774, 75, 76sylanbrc 581 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → 𝑧(𝑋 × 𝑌)𝑤)
78 brin 5205 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧((𝑊𝑋) ∩ (𝑋 × 𝑌))𝑤 ↔ (𝑧(𝑊𝑋)𝑤𝑧(𝑋 × 𝑌)𝑤))
7978rbaib 537 . . . . . . . . . . . . . . . . . . . . . . . . 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 278 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → (𝑧𝑅𝑤𝑧(𝑊𝑋)𝑤))
821, 2fpwwe2lem2 10675 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑 → (𝑌𝑊𝑅 ↔ ((𝑌𝐴𝑅 ⊆ (𝑌 × 𝑌)) ∧ (𝑅 We 𝑌 ∧ ∀𝑦𝑌 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))))
8382biimpa 475 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑌𝑊𝑅) → ((𝑌𝐴𝑅 ⊆ (𝑌 × 𝑌)) ∧ (𝑅 We 𝑌 ∧ ∀𝑦𝑌 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦)))
8483adantrr 715 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → ((𝑌𝐴𝑅 ⊆ (𝑌 × 𝑌)) ∧ (𝑅 We 𝑌 ∧ ∀𝑦𝑌 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦)))
8584simpld 493 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → (𝑌𝐴𝑅 ⊆ (𝑌 × 𝑌)))
8685simprd 494 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑅 ⊆ (𝑌 × 𝑌))
8786ad5ant12 754 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → 𝑅 ⊆ (𝑌 × 𝑌))
8887ssbrd 5196 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → (𝑧𝑅𝑤𝑧(𝑌 × 𝑌)𝑤))
89 brxp 5731 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧(𝑌 × 𝑌)𝑤 ↔ (𝑧𝑌𝑤𝑌))
9089simplbi 496 . . . . . . . . . . . . . . . . . . . . . . . 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 259 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → (𝑧(𝑊𝑋)𝑤𝑧𝑌))
9368, 92mtod 197 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → ¬ 𝑧(𝑊𝑋)𝑤)
9431ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → (𝑊𝑋) We 𝑋)
95 weso 5673 . . . . . . . . . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑌𝑋)
9897sselda 3979 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → 𝑤𝑋)
99 sotric 5622 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑊𝑋) Or 𝑋 ∧ (𝑤𝑋𝑧𝑋)) → (𝑤(𝑊𝑋)𝑧 ↔ ¬ (𝑤 = 𝑧𝑧(𝑊𝑋)𝑤)))
100 ioran 981 . . . . . . . . . . . . . . . . . . . . . . 23 (¬ (𝑤 = 𝑧𝑧(𝑊𝑋)𝑤) ↔ (¬ 𝑤 = 𝑧 ∧ ¬ 𝑧(𝑊𝑋)𝑤))
10199, 100bitrdi 286 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑊𝑋) Or 𝑋 ∧ (𝑤𝑋𝑧𝑋)) → (𝑤(𝑊𝑋)𝑧 ↔ (¬ 𝑤 = 𝑧 ∧ ¬ 𝑧(𝑊𝑋)𝑤)))
10296, 98, 74, 101syl12anc 835 . . . . . . . . . . . . . . . . . . . . 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 711 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → 𝑤(𝑊𝑋)𝑧)
104103, 44sylibr 233 . . . . . . . . . . . . . . . . . . 19 (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤𝑌) → 𝑤 ∈ ((𝑊𝑋) “ {𝑧}))
105104ex 411 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑤𝑌𝑤 ∈ ((𝑊𝑋) “ {𝑧})))
106105ssrdv 3985 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑌 ⊆ ((𝑊𝑋) “ {𝑧}))
107 simprr 771 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)
108106, 107eqssd 3997 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑌 = ((𝑊𝑋) “ {𝑧}))
109 in32 4223 . . . . . . . . . . . . . . . . . 18 (((𝑊𝑋) ∩ (𝑋 × 𝑌)) ∩ (𝑌 × 𝑌)) = (((𝑊𝑋) ∩ (𝑌 × 𝑌)) ∩ (𝑋 × 𝑌))
110 simplrr 776 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))
111110ineq1d 4212 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑅 ∩ (𝑌 × 𝑌)) = (((𝑊𝑋) ∩ (𝑋 × 𝑌)) ∩ (𝑌 × 𝑌)))
11286ad2antrr 724 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑅 ⊆ (𝑌 × 𝑌))
113 dfss2 3965 . . . . . . . . . . . . . . . . . . . 20 (𝑅 ⊆ (𝑌 × 𝑌) ↔ (𝑅 ∩ (𝑌 × 𝑌)) = 𝑅)
114112, 113sylib 217 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑅 ∩ (𝑌 × 𝑌)) = 𝑅)
115111, 114eqtr3d 2768 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (((𝑊𝑋) ∩ (𝑋 × 𝑌)) ∩ (𝑌 × 𝑌)) = 𝑅)
116 inss2 4231 . . . . . . . . . . . . . . . . . . . 20 ((𝑊𝑋) ∩ (𝑌 × 𝑌)) ⊆ (𝑌 × 𝑌)
117 xpss1 5701 . . . . . . . . . . . . . . . . . . . . 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 3991 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → ((𝑊𝑋) ∩ (𝑌 × 𝑌)) ⊆ (𝑋 × 𝑌))
120 dfss2 3965 . . . . . . . . . . . . . . . . . . 19 (((𝑊𝑋) ∩ (𝑌 × 𝑌)) ⊆ (𝑋 × 𝑌) ↔ (((𝑊𝑋) ∩ (𝑌 × 𝑌)) ∩ (𝑋 × 𝑌)) = ((𝑊𝑋) ∩ (𝑌 × 𝑌)))
121119, 120sylib 217 . . . . . . . . . . . . . . . . . 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 2790 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑅 = ((𝑊𝑋) ∩ (𝑌 × 𝑌)))
123108sqxpeqd 5714 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑌 × 𝑌) = (((𝑊𝑋) “ {𝑧}) × ((𝑊𝑋) “ {𝑧})))
124123ineq2d 4213 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → ((𝑊𝑋) ∩ (𝑌 × 𝑌)) = ((𝑊𝑋) ∩ (((𝑊𝑋) “ {𝑧}) × ((𝑊𝑋) “ {𝑧}))))
125122, 124eqtrd 2766 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑅 = ((𝑊𝑋) ∩ (((𝑊𝑋) “ {𝑧}) × ((𝑊𝑋) “ {𝑧}))))
126108, 125oveq12d 7442 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑌𝐹𝑅) = (((𝑊𝑋) “ {𝑧})𝐹((𝑊𝑋) ∩ (((𝑊𝑋) “ {𝑧}) × ((𝑊𝑋) “ {𝑧})))))
12718adantr 479 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝐴𝑉)
12822adantr 479 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑋𝑊(𝑊𝑋))
129128ad2antrr 724 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑋𝑊(𝑊𝑋))
1301, 127, 129fpwwe2lem3 10676 . . . . . . . . . . . . . . . 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 685 . . . . . . . . . . . . . . 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 2766 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑌𝐹𝑅) = 𝑧)
133132, 62eqneltrd 2846 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋𝑌) ∧ ((𝑊𝑋) “ {𝑧}) ⊆ 𝑌)) → ¬ (𝑌𝐹𝑅) ∈ 𝑌)
134133rexlimdvaa 3146 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → (∃𝑧 ∈ (𝑋𝑌)((𝑊𝑋) “ {𝑧}) ⊆ 𝑌 → ¬ (𝑌𝐹𝑅) ∈ 𝑌))
13560, 134sylbid 239 . . . . . . . . . . 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 2949 . . . . . . . . 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 4366 . . . . . . . 8 (𝑋𝑌 ↔ (𝑋𝑌) = ∅)
140138, 139sylibr 233 . . . . . . 7 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))) → 𝑋𝑌)
141140ex 411 . . . . . 6 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → ((𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌))) → 𝑋𝑌))
1423adantlr 713 . . . . . . 7 (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
143 simprl 769 . . . . . . 7 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑌𝑊𝑅)
1441, 17, 142, 128, 143fpwwe2lem9 10682 . . . . . 6 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → ((𝑋𝑌 ∧ (𝑊𝑋) = (𝑅 ∩ (𝑌 × 𝑋))) ∨ (𝑌𝑋𝑅 = ((𝑊𝑋) ∩ (𝑋 × 𝑌)))))
14515, 141, 144mpjaod 858 . . . . 5 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑋𝑌)
14613, 145eqssd 3997 . . . 4 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑌 = 𝑋)
1476adantr 479 . . . . . 6 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → Fun 𝑊)
148146, 143eqbrtrrd 5177 . . . . . 6 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑋𝑊𝑅)
149 funbrfv 6952 . . . . . 6 (Fun 𝑊 → (𝑋𝑊𝑅 → (𝑊𝑋) = 𝑅))
150147, 148, 149sylc 65 . . . . 5 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → (𝑊𝑋) = 𝑅)
151150eqcomd 2732 . . . 4 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑅 = (𝑊𝑋))
152146, 151jca 510 . . 3 ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → (𝑌 = 𝑋𝑅 = (𝑊𝑋)))
153152ex 411 . 2 (𝜑 → ((𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌) → (𝑌 = 𝑋𝑅 = (𝑊𝑋))))
1541, 2, 3, 4fpwwe2lem12 10685 . . . 4 (𝜑 → (𝑋𝐹(𝑊𝑋)) ∈ 𝑋)
15522, 154jca 510 . . 3 (𝜑 → (𝑋𝑊(𝑊𝑋) ∧ (𝑋𝐹(𝑊𝑋)) ∈ 𝑋))
156 breq12 5158 . . . 4 ((𝑌 = 𝑋𝑅 = (𝑊𝑋)) → (𝑌𝑊𝑅𝑋𝑊(𝑊𝑋)))
157 oveq12 7433 . . . . 5 ((𝑌 = 𝑋𝑅 = (𝑊𝑋)) → (𝑌𝐹𝑅) = (𝑋𝐹(𝑊𝑋)))
158 simpl 481 . . . . 5 ((𝑌 = 𝑋𝑅 = (𝑊𝑋)) → 𝑌 = 𝑋)
159157, 158eleq12d 2820 . . . 4 ((𝑌 = 𝑋𝑅 = (𝑊𝑋)) → ((𝑌𝐹𝑅) ∈ 𝑌 ↔ (𝑋𝐹(𝑊𝑋)) ∈ 𝑋))
160156, 159anbi12d 630 . . 3 ((𝑌 = 𝑋𝑅 = (𝑊𝑋)) → ((𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌) ↔ (𝑋𝑊(𝑊𝑋) ∧ (𝑋𝐹(𝑊𝑋)) ∈ 𝑋)))
161155, 160syl5ibrcom 246 . 2 (𝜑 → ((𝑌 = 𝑋𝑅 = (𝑊𝑋)) → (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)))
162153, 161impbid 211 1 (𝜑 → ((𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌) ↔ (𝑌 = 𝑋𝑅 = (𝑊𝑋))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wo 845  w3a 1084   = wceq 1534  wcel 2099  wne 2930  wral 3051  wrex 3060  Vcvv 3462  [wsbc 3776  cdif 3944  cin 3946  wss 3947  c0 4325  𝒫 cpw 4607  {csn 4633   cuni 4913   class class class wbr 5153  {copab 5215   Or wor 5593   Fr wfr 5634   We wwe 5636   × cxp 5680  ccnv 5681  dom cdm 5682  cima 5685  Fun wfun 6548  cfv 6554  (class class class)co 7424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-tp 4638  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-isom 6563  df-riota 7380  df-ov 7427  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-oi 9553
This theorem is referenced by:  fpwwe  10689  canthwelem  10693  pwfseqlem4  10705
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