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Theorem fpwwecbv 10682
Description: Lemma for fpwwe 10684. (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 482 . . . . . 6 ((𝑥 = 𝑎𝑟 = 𝑠) → 𝑥 = 𝑎)
32sseq1d 4027 . . . . 5 ((𝑥 = 𝑎𝑟 = 𝑠) → (𝑥𝐴𝑎𝐴))
4 simpr 484 . . . . . 6 ((𝑥 = 𝑎𝑟 = 𝑠) → 𝑟 = 𝑠)
52sqxpeqd 5721 . . . . . 6 ((𝑥 = 𝑎𝑟 = 𝑠) → (𝑥 × 𝑥) = (𝑎 × 𝑎))
64, 5sseq12d 4029 . . . . 5 ((𝑥 = 𝑎𝑟 = 𝑠) → (𝑟 ⊆ (𝑥 × 𝑥) ↔ 𝑠 ⊆ (𝑎 × 𝑎)))
73, 6anbi12d 632 . . . 4 ((𝑥 = 𝑎𝑟 = 𝑠) → ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ↔ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎))))
84, 2weeq12d 5678 . . . . 5 ((𝑥 = 𝑎𝑟 = 𝑠) → (𝑟 We 𝑥𝑠 We 𝑎))
9 sneq 4641 . . . . . . . . . 10 (𝑦 = 𝑧 → {𝑦} = {𝑧})
109imaeq2d 6080 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑟 “ {𝑦}) = (𝑟 “ {𝑧}))
1110fveq2d 6911 . . . . . . . 8 (𝑦 = 𝑧 → (𝐹‘(𝑟 “ {𝑦})) = (𝐹‘(𝑟 “ {𝑧})))
12 id 22 . . . . . . . 8 (𝑦 = 𝑧𝑦 = 𝑧)
1311, 12eqeq12d 2751 . . . . . . 7 (𝑦 = 𝑧 → ((𝐹‘(𝑟 “ {𝑦})) = 𝑦 ↔ (𝐹‘(𝑟 “ {𝑧})) = 𝑧))
1413cbvralvw 3235 . . . . . 6 (∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦 ↔ ∀𝑧𝑥 (𝐹‘(𝑟 “ {𝑧})) = 𝑧)
154cnveqd 5889 . . . . . . . . 9 ((𝑥 = 𝑎𝑟 = 𝑠) → 𝑟 = 𝑠)
1615imaeq1d 6079 . . . . . . . 8 ((𝑥 = 𝑎𝑟 = 𝑠) → (𝑟 “ {𝑧}) = (𝑠 “ {𝑧}))
1716fveqeq2d 6915 . . . . . . 7 ((𝑥 = 𝑎𝑟 = 𝑠) → ((𝐹‘(𝑟 “ {𝑧})) = 𝑧 ↔ (𝐹‘(𝑠 “ {𝑧})) = 𝑧))
182, 17raleqbidv 3344 . . . . . 6 ((𝑥 = 𝑎𝑟 = 𝑠) → (∀𝑧𝑥 (𝐹‘(𝑟 “ {𝑧})) = 𝑧 ↔ ∀𝑧𝑎 (𝐹‘(𝑠 “ {𝑧})) = 𝑧))
1914, 18bitrid 283 . . . . 5 ((𝑥 = 𝑎𝑟 = 𝑠) → (∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦 ↔ ∀𝑧𝑎 (𝐹‘(𝑠 “ {𝑧})) = 𝑧))
208, 19anbi12d 632 . . . 4 ((𝑥 = 𝑎𝑟 = 𝑠) → ((𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦) ↔ (𝑠 We 𝑎 ∧ ∀𝑧𝑎 (𝐹‘(𝑠 “ {𝑧})) = 𝑧)))
217, 20anbi12d 632 . . 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))))
2221cbvopabv 5221 . 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))}
231, 22eqtri 2763 1 𝑊 = {⟨𝑎, 𝑠⟩ ∣ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧𝑎 (𝐹‘(𝑠 “ {𝑧})) = 𝑧))}
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1537  wral 3059  wss 3963  {csn 4631  {copab 5210   We wwe 5640   × cxp 5687  ccnv 5688  cima 5692  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fv 6571
This theorem is referenced by:  canthnum  10687  canthp1  10692
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