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Theorem fpwwecbv 10054
Description: Lemma for fpwwe 10056. (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 483 . . . . . 6 ((𝑥 = 𝑎𝑟 = 𝑠) → 𝑥 = 𝑎)
32sseq1d 3995 . . . . 5 ((𝑥 = 𝑎𝑟 = 𝑠) → (𝑥𝐴𝑎𝐴))
4 simpr 485 . . . . . 6 ((𝑥 = 𝑎𝑟 = 𝑠) → 𝑟 = 𝑠)
52sqxpeqd 5580 . . . . . 6 ((𝑥 = 𝑎𝑟 = 𝑠) → (𝑥 × 𝑥) = (𝑎 × 𝑎))
64, 5sseq12d 3997 . . . . 5 ((𝑥 = 𝑎𝑟 = 𝑠) → (𝑟 ⊆ (𝑥 × 𝑥) ↔ 𝑠 ⊆ (𝑎 × 𝑎)))
73, 6anbi12d 630 . . . 4 ((𝑥 = 𝑎𝑟 = 𝑠) → ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ↔ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎))))
8 weeq2 5537 . . . . . 6 (𝑥 = 𝑎 → (𝑟 We 𝑥𝑟 We 𝑎))
9 weeq1 5536 . . . . . 6 (𝑟 = 𝑠 → (𝑟 We 𝑎𝑠 We 𝑎))
108, 9sylan9bb 510 . . . . 5 ((𝑥 = 𝑎𝑟 = 𝑠) → (𝑟 We 𝑥𝑠 We 𝑎))
11 sneq 4567 . . . . . . . . . 10 (𝑦 = 𝑧 → {𝑦} = {𝑧})
1211imaeq2d 5922 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑟 “ {𝑦}) = (𝑟 “ {𝑧}))
1312fveq2d 6667 . . . . . . . 8 (𝑦 = 𝑧 → (𝐹‘(𝑟 “ {𝑦})) = (𝐹‘(𝑟 “ {𝑧})))
14 id 22 . . . . . . . 8 (𝑦 = 𝑧𝑦 = 𝑧)
1513, 14eqeq12d 2834 . . . . . . 7 (𝑦 = 𝑧 → ((𝐹‘(𝑟 “ {𝑦})) = 𝑦 ↔ (𝐹‘(𝑟 “ {𝑧})) = 𝑧))
1615cbvralvw 3447 . . . . . 6 (∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦 ↔ ∀𝑧𝑥 (𝐹‘(𝑟 “ {𝑧})) = 𝑧)
174cnveqd 5739 . . . . . . . . 9 ((𝑥 = 𝑎𝑟 = 𝑠) → 𝑟 = 𝑠)
1817imaeq1d 5921 . . . . . . . 8 ((𝑥 = 𝑎𝑟 = 𝑠) → (𝑟 “ {𝑧}) = (𝑠 “ {𝑧}))
1918fveqeq2d 6671 . . . . . . 7 ((𝑥 = 𝑎𝑟 = 𝑠) → ((𝐹‘(𝑟 “ {𝑧})) = 𝑧 ↔ (𝐹‘(𝑠 “ {𝑧})) = 𝑧))
202, 19raleqbidv 3399 . . . . . 6 ((𝑥 = 𝑎𝑟 = 𝑠) → (∀𝑧𝑥 (𝐹‘(𝑟 “ {𝑧})) = 𝑧 ↔ ∀𝑧𝑎 (𝐹‘(𝑠 “ {𝑧})) = 𝑧))
2116, 20syl5bb 284 . . . . 5 ((𝑥 = 𝑎𝑟 = 𝑠) → (∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦 ↔ ∀𝑧𝑎 (𝐹‘(𝑠 “ {𝑧})) = 𝑧))
2210, 21anbi12d 630 . . . 4 ((𝑥 = 𝑎𝑟 = 𝑠) → ((𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦) ↔ (𝑠 We 𝑎 ∧ ∀𝑧𝑎 (𝐹‘(𝑠 “ {𝑧})) = 𝑧)))
237, 22anbi12d 630 . . 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))))
2423cbvopabv 5129 . 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))}
251, 24eqtri 2841 1 𝑊 = {⟨𝑎, 𝑠⟩ ∣ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧𝑎 (𝐹‘(𝑠 “ {𝑧})) = 𝑧))}
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1528  wral 3135  wss 3933  {csn 4557  {copab 5119   We wwe 5506   × cxp 5546  ccnv 5547  cima 5551  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fv 6356
This theorem is referenced by:  canthnum  10059  canthp1  10064
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