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Theorem fpwwe2cbv 9704
Description: Lemma for fpwwe2 9717. (Contributed by Mario Carneiro, 3-Jun-2015.)
Hypothesis
Ref Expression
fpwwe2.1 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
Assertion
Ref Expression
fpwwe2cbv 𝑊 = {⟨𝑎, 𝑠⟩ ∣ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧𝑎 [(𝑠 “ {𝑧}) / 𝑣](𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))}
Distinct variable groups:   𝑦,𝑢   𝑟,𝑎,𝑠,𝑢,𝑣,𝑥,𝑦,𝑧,𝐹   𝐴,𝑎,𝑟,𝑠,𝑥,𝑧
Allowed substitution hints:   𝐴(𝑦,𝑣,𝑢)   𝑊(𝑥,𝑦,𝑧,𝑣,𝑢,𝑠,𝑟,𝑎)

Proof of Theorem fpwwe2cbv
StepHypRef Expression
1 fpwwe2.1 . 2 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
2 simpl 474 . . . . . 6 ((𝑥 = 𝑎𝑟 = 𝑠) → 𝑥 = 𝑎)
32sseq1d 3791 . . . . 5 ((𝑥 = 𝑎𝑟 = 𝑠) → (𝑥𝐴𝑎𝐴))
4 simpr 477 . . . . . 6 ((𝑥 = 𝑎𝑟 = 𝑠) → 𝑟 = 𝑠)
52sqxpeqd 5308 . . . . . 6 ((𝑥 = 𝑎𝑟 = 𝑠) → (𝑥 × 𝑥) = (𝑎 × 𝑎))
64, 5sseq12d 3793 . . . . 5 ((𝑥 = 𝑎𝑟 = 𝑠) → (𝑟 ⊆ (𝑥 × 𝑥) ↔ 𝑠 ⊆ (𝑎 × 𝑎)))
73, 6anbi12d 624 . . . 4 ((𝑥 = 𝑎𝑟 = 𝑠) → ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ↔ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎))))
8 weeq2 5265 . . . . . 6 (𝑥 = 𝑎 → (𝑟 We 𝑥𝑟 We 𝑎))
9 weeq1 5264 . . . . . 6 (𝑟 = 𝑠 → (𝑟 We 𝑎𝑠 We 𝑎))
108, 9sylan9bb 505 . . . . 5 ((𝑥 = 𝑎𝑟 = 𝑠) → (𝑟 We 𝑥𝑠 We 𝑎))
11 id 22 . . . . . . . . . . 11 (𝑢 = 𝑣𝑢 = 𝑣)
1211sqxpeqd 5308 . . . . . . . . . . . 12 (𝑢 = 𝑣 → (𝑢 × 𝑢) = (𝑣 × 𝑣))
1312ineq2d 3975 . . . . . . . . . . 11 (𝑢 = 𝑣 → (𝑟 ∩ (𝑢 × 𝑢)) = (𝑟 ∩ (𝑣 × 𝑣)))
1411, 13oveq12d 6859 . . . . . . . . . 10 (𝑢 = 𝑣 → (𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = (𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))))
1514eqeq1d 2766 . . . . . . . . 9 (𝑢 = 𝑣 → ((𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ (𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑦))
1615cbvsbcv 3625 . . . . . . . 8 ([(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦[(𝑟 “ {𝑦}) / 𝑣](𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑦)
17 sneq 4343 . . . . . . . . . 10 (𝑦 = 𝑧 → {𝑦} = {𝑧})
1817imaeq2d 5647 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑟 “ {𝑦}) = (𝑟 “ {𝑧}))
19 eqeq2 2775 . . . . . . . . 9 (𝑦 = 𝑧 → ((𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑦 ↔ (𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑧))
2018, 19sbceqbid 3602 . . . . . . . 8 (𝑦 = 𝑧 → ([(𝑟 “ {𝑦}) / 𝑣](𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑦[(𝑟 “ {𝑧}) / 𝑣](𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑧))
2116, 20syl5bb 274 . . . . . . 7 (𝑦 = 𝑧 → ([(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦[(𝑟 “ {𝑧}) / 𝑣](𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑧))
2221cbvralv 3318 . . . . . 6 (∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ ∀𝑧𝑥 [(𝑟 “ {𝑧}) / 𝑣](𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑧)
234cnveqd 5465 . . . . . . . . 9 ((𝑥 = 𝑎𝑟 = 𝑠) → 𝑟 = 𝑠)
2423imaeq1d 5646 . . . . . . . 8 ((𝑥 = 𝑎𝑟 = 𝑠) → (𝑟 “ {𝑧}) = (𝑠 “ {𝑧}))
254ineq1d 3974 . . . . . . . . . 10 ((𝑥 = 𝑎𝑟 = 𝑠) → (𝑟 ∩ (𝑣 × 𝑣)) = (𝑠 ∩ (𝑣 × 𝑣)))
2625oveq2d 6857 . . . . . . . . 9 ((𝑥 = 𝑎𝑟 = 𝑠) → (𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = (𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))))
2726eqeq1d 2766 . . . . . . . 8 ((𝑥 = 𝑎𝑟 = 𝑠) → ((𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑧 ↔ (𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))
2824, 27sbceqbid 3602 . . . . . . 7 ((𝑥 = 𝑎𝑟 = 𝑠) → ([(𝑟 “ {𝑧}) / 𝑣](𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑧[(𝑠 “ {𝑧}) / 𝑣](𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))
292, 28raleqbidv 3299 . . . . . 6 ((𝑥 = 𝑎𝑟 = 𝑠) → (∀𝑧𝑥 [(𝑟 “ {𝑧}) / 𝑣](𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑧 ↔ ∀𝑧𝑎 [(𝑠 “ {𝑧}) / 𝑣](𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))
3022, 29syl5bb 274 . . . . 5 ((𝑥 = 𝑎𝑟 = 𝑠) → (∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ ∀𝑧𝑎 [(𝑠 “ {𝑧}) / 𝑣](𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))
3110, 30anbi12d 624 . . . 4 ((𝑥 = 𝑎𝑟 = 𝑠) → ((𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦) ↔ (𝑠 We 𝑎 ∧ ∀𝑧𝑎 [(𝑠 “ {𝑧}) / 𝑣](𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧)))
327, 31anbi12d 624 . . 3 ((𝑥 = 𝑎𝑟 = 𝑠) → (((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦)) ↔ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧𝑎 [(𝑠 “ {𝑧}) / 𝑣](𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))))
3332cbvopabv 4880 . 2 {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))} = {⟨𝑎, 𝑠⟩ ∣ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧𝑎 [(𝑠 “ {𝑧}) / 𝑣](𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))}
341, 33eqtri 2786 1 𝑊 = {⟨𝑎, 𝑠⟩ ∣ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧𝑎 [(𝑠 “ {𝑧}) / 𝑣](𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))}
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1652  wral 3054  [wsbc 3595  cin 3730  wss 3731  {csn 4333  {copab 4870   We wwe 5234   × cxp 5274  ccnv 5275  cima 5279  (class class class)co 6841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3351  df-sbc 3596  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-nul 4079  df-if 4243  df-sn 4334  df-pr 4336  df-op 4340  df-uni 4594  df-br 4809  df-opab 4871  df-po 5197  df-so 5198  df-fr 5235  df-we 5237  df-xp 5282  df-cnv 5284  df-dm 5286  df-rn 5287  df-res 5288  df-ima 5289  df-iota 6030  df-fv 6075  df-ov 6844
This theorem is referenced by:  fpwwe2lem12  9715  fpwwe2lem13  9716  canthwe  9725  pwfseqlem5  9737
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