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Theorem fpwwecbv 9504
Description: Lemma for fpwwe 9506. (Contributed by Mario Carneiro, 15-May-2015.)
Hypothesis
Ref Expression
fpwwe.1 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦))}
Assertion
Ref Expression
fpwwecbv 𝑊 = {⟨𝑎, 𝑠⟩ ∣ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧𝑎 (𝐹‘(𝑠 “ {𝑧})) = 𝑧))}
Distinct variable groups:   𝑟,𝑎,𝑠,𝑥,𝐴   𝑦,𝑎,𝑧,𝐹,𝑟,𝑠,𝑥
Allowed substitution hints:   𝐴(𝑦,𝑧)   𝑊(𝑥,𝑦,𝑧,𝑠,𝑟,𝑎)

Proof of Theorem fpwwecbv
StepHypRef Expression
1 fpwwe.1 . 2 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦))}
2 simpl 472 . . . . . 6 ((𝑥 = 𝑎𝑟 = 𝑠) → 𝑥 = 𝑎)
32sseq1d 3665 . . . . 5 ((𝑥 = 𝑎𝑟 = 𝑠) → (𝑥𝐴𝑎𝐴))
4 simpr 476 . . . . . 6 ((𝑥 = 𝑎𝑟 = 𝑠) → 𝑟 = 𝑠)
52sqxpeqd 5175 . . . . . 6 ((𝑥 = 𝑎𝑟 = 𝑠) → (𝑥 × 𝑥) = (𝑎 × 𝑎))
64, 5sseq12d 3667 . . . . 5 ((𝑥 = 𝑎𝑟 = 𝑠) → (𝑟 ⊆ (𝑥 × 𝑥) ↔ 𝑠 ⊆ (𝑎 × 𝑎)))
73, 6anbi12d 747 . . . 4 ((𝑥 = 𝑎𝑟 = 𝑠) → ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ↔ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎))))
8 weeq2 5132 . . . . . 6 (𝑥 = 𝑎 → (𝑟 We 𝑥𝑟 We 𝑎))
9 weeq1 5131 . . . . . 6 (𝑟 = 𝑠 → (𝑟 We 𝑎𝑠 We 𝑎))
108, 9sylan9bb 736 . . . . 5 ((𝑥 = 𝑎𝑟 = 𝑠) → (𝑟 We 𝑥𝑠 We 𝑎))
11 sneq 4220 . . . . . . . . . 10 (𝑦 = 𝑧 → {𝑦} = {𝑧})
1211imaeq2d 5501 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑟 “ {𝑦}) = (𝑟 “ {𝑧}))
1312fveq2d 6233 . . . . . . . 8 (𝑦 = 𝑧 → (𝐹‘(𝑟 “ {𝑦})) = (𝐹‘(𝑟 “ {𝑧})))
14 id 22 . . . . . . . 8 (𝑦 = 𝑧𝑦 = 𝑧)
1513, 14eqeq12d 2666 . . . . . . 7 (𝑦 = 𝑧 → ((𝐹‘(𝑟 “ {𝑦})) = 𝑦 ↔ (𝐹‘(𝑟 “ {𝑧})) = 𝑧))
1615cbvralv 3201 . . . . . 6 (∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦 ↔ ∀𝑧𝑥 (𝐹‘(𝑟 “ {𝑧})) = 𝑧)
174cnveqd 5330 . . . . . . . . . 10 ((𝑥 = 𝑎𝑟 = 𝑠) → 𝑟 = 𝑠)
1817imaeq1d 5500 . . . . . . . . 9 ((𝑥 = 𝑎𝑟 = 𝑠) → (𝑟 “ {𝑧}) = (𝑠 “ {𝑧}))
1918fveq2d 6233 . . . . . . . 8 ((𝑥 = 𝑎𝑟 = 𝑠) → (𝐹‘(𝑟 “ {𝑧})) = (𝐹‘(𝑠 “ {𝑧})))
2019eqeq1d 2653 . . . . . . 7 ((𝑥 = 𝑎𝑟 = 𝑠) → ((𝐹‘(𝑟 “ {𝑧})) = 𝑧 ↔ (𝐹‘(𝑠 “ {𝑧})) = 𝑧))
212, 20raleqbidv 3182 . . . . . 6 ((𝑥 = 𝑎𝑟 = 𝑠) → (∀𝑧𝑥 (𝐹‘(𝑟 “ {𝑧})) = 𝑧 ↔ ∀𝑧𝑎 (𝐹‘(𝑠 “ {𝑧})) = 𝑧))
2216, 21syl5bb 272 . . . . 5 ((𝑥 = 𝑎𝑟 = 𝑠) → (∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦 ↔ ∀𝑧𝑎 (𝐹‘(𝑠 “ {𝑧})) = 𝑧))
2310, 22anbi12d 747 . . . 4 ((𝑥 = 𝑎𝑟 = 𝑠) → ((𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦) ↔ (𝑠 We 𝑎 ∧ ∀𝑧𝑎 (𝐹‘(𝑠 “ {𝑧})) = 𝑧)))
247, 23anbi12d 747 . . 3 ((𝑥 = 𝑎𝑟 = 𝑠) → (((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦)) ↔ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧𝑎 (𝐹‘(𝑠 “ {𝑧})) = 𝑧))))
2524cbvopabv 4755 . 2 {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦))} = {⟨𝑎, 𝑠⟩ ∣ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧𝑎 (𝐹‘(𝑠 “ {𝑧})) = 𝑧))}
261, 25eqtri 2673 1 𝑊 = {⟨𝑎, 𝑠⟩ ∣ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧𝑎 (𝐹‘(𝑠 “ {𝑧})) = 𝑧))}
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1523  wral 2941  wss 3607  {csn 4210  {copab 4745   We wwe 5101   × cxp 5141  ccnv 5142  cima 5146  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fv 5934
This theorem is referenced by:  canthnum  9509  canthp1  9514
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