MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fpwwecbv Structured version   Visualization version   GIF version

Theorem fpwwecbv 10401
Description: Lemma for fpwwe 10403. (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 3957 . . . . 5 ((𝑥 = 𝑎𝑟 = 𝑠) → (𝑥𝐴𝑎𝐴))
4 simpr 485 . . . . . 6 ((𝑥 = 𝑎𝑟 = 𝑠) → 𝑟 = 𝑠)
52sqxpeqd 5622 . . . . . 6 ((𝑥 = 𝑎𝑟 = 𝑠) → (𝑥 × 𝑥) = (𝑎 × 𝑎))
64, 5sseq12d 3959 . . . . 5 ((𝑥 = 𝑎𝑟 = 𝑠) → (𝑟 ⊆ (𝑥 × 𝑥) ↔ 𝑠 ⊆ (𝑎 × 𝑎)))
73, 6anbi12d 631 . . . 4 ((𝑥 = 𝑎𝑟 = 𝑠) → ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ↔ (𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎))))
8 weeq2 5579 . . . . . 6 (𝑥 = 𝑎 → (𝑟 We 𝑥𝑟 We 𝑎))
9 weeq1 5578 . . . . . 6 (𝑟 = 𝑠 → (𝑟 We 𝑎𝑠 We 𝑎))
108, 9sylan9bb 510 . . . . 5 ((𝑥 = 𝑎𝑟 = 𝑠) → (𝑟 We 𝑥𝑠 We 𝑎))
11 sneq 4577 . . . . . . . . . 10 (𝑦 = 𝑧 → {𝑦} = {𝑧})
1211imaeq2d 5968 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑟 “ {𝑦}) = (𝑟 “ {𝑧}))
1312fveq2d 6775 . . . . . . . 8 (𝑦 = 𝑧 → (𝐹‘(𝑟 “ {𝑦})) = (𝐹‘(𝑟 “ {𝑧})))
14 id 22 . . . . . . . 8 (𝑦 = 𝑧𝑦 = 𝑧)
1513, 14eqeq12d 2756 . . . . . . 7 (𝑦 = 𝑧 → ((𝐹‘(𝑟 “ {𝑦})) = 𝑦 ↔ (𝐹‘(𝑟 “ {𝑧})) = 𝑧))
1615cbvralvw 3381 . . . . . 6 (∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦 ↔ ∀𝑧𝑥 (𝐹‘(𝑟 “ {𝑧})) = 𝑧)
174cnveqd 5783 . . . . . . . . 9 ((𝑥 = 𝑎𝑟 = 𝑠) → 𝑟 = 𝑠)
1817imaeq1d 5967 . . . . . . . 8 ((𝑥 = 𝑎𝑟 = 𝑠) → (𝑟 “ {𝑧}) = (𝑠 “ {𝑧}))
1918fveqeq2d 6779 . . . . . . 7 ((𝑥 = 𝑎𝑟 = 𝑠) → ((𝐹‘(𝑟 “ {𝑧})) = 𝑧 ↔ (𝐹‘(𝑠 “ {𝑧})) = 𝑧))
202, 19raleqbidv 3335 . . . . . 6 ((𝑥 = 𝑎𝑟 = 𝑠) → (∀𝑧𝑥 (𝐹‘(𝑟 “ {𝑧})) = 𝑧 ↔ ∀𝑧𝑎 (𝐹‘(𝑠 “ {𝑧})) = 𝑧))
2116, 20bitrid 282 . . . . 5 ((𝑥 = 𝑎𝑟 = 𝑠) → (∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦 ↔ ∀𝑧𝑎 (𝐹‘(𝑠 “ {𝑧})) = 𝑧))
2210, 21anbi12d 631 . . . 4 ((𝑥 = 𝑎𝑟 = 𝑠) → ((𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦) ↔ (𝑠 We 𝑎 ∧ ∀𝑧𝑎 (𝐹‘(𝑠 “ {𝑧})) = 𝑧)))
237, 22anbi12d 631 . . 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 5152 . 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 2768 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 1542  wral 3066  wss 3892  {csn 4567  {copab 5141   We wwe 5544   × cxp 5588  ccnv 5589  cima 5593  cfv 6432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-cnv 5598  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6390  df-fv 6440
This theorem is referenced by:  canthnum  10406  canthp1  10411
  Copyright terms: Public domain W3C validator