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

Theorem fpwwecbv 9788
Description: Lemma for fpwwe 9790. (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 476 . . . . . 6 ((𝑥 = 𝑎𝑟 = 𝑠) → 𝑥 = 𝑎)
32sseq1d 3857 . . . . 5 ((𝑥 = 𝑎𝑟 = 𝑠) → (𝑥𝐴𝑎𝐴))
4 simpr 479 . . . . . 6 ((𝑥 = 𝑎𝑟 = 𝑠) → 𝑟 = 𝑠)
52sqxpeqd 5378 . . . . . 6 ((𝑥 = 𝑎𝑟 = 𝑠) → (𝑥 × 𝑥) = (𝑎 × 𝑎))
64, 5sseq12d 3859 . . . . 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 5335 . . . . . 6 (𝑥 = 𝑎 → (𝑟 We 𝑥𝑟 We 𝑎))
9 weeq1 5334 . . . . . 6 (𝑟 = 𝑠 → (𝑟 We 𝑎𝑠 We 𝑎))
108, 9sylan9bb 505 . . . . 5 ((𝑥 = 𝑎𝑟 = 𝑠) → (𝑟 We 𝑥𝑠 We 𝑎))
11 sneq 4409 . . . . . . . . . 10 (𝑦 = 𝑧 → {𝑦} = {𝑧})
1211imaeq2d 5711 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑟 “ {𝑦}) = (𝑟 “ {𝑧}))
1312fveq2d 6441 . . . . . . . 8 (𝑦 = 𝑧 → (𝐹‘(𝑟 “ {𝑦})) = (𝐹‘(𝑟 “ {𝑧})))
14 id 22 . . . . . . . 8 (𝑦 = 𝑧𝑦 = 𝑧)
1513, 14eqeq12d 2840 . . . . . . 7 (𝑦 = 𝑧 → ((𝐹‘(𝑟 “ {𝑦})) = 𝑦 ↔ (𝐹‘(𝑟 “ {𝑧})) = 𝑧))
1615cbvralv 3383 . . . . . 6 (∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦 ↔ ∀𝑧𝑥 (𝐹‘(𝑟 “ {𝑧})) = 𝑧)
174cnveqd 5534 . . . . . . . . 9 ((𝑥 = 𝑎𝑟 = 𝑠) → 𝑟 = 𝑠)
1817imaeq1d 5710 . . . . . . . 8 ((𝑥 = 𝑎𝑟 = 𝑠) → (𝑟 “ {𝑧}) = (𝑠 “ {𝑧}))
1918fveqeq2d 6445 . . . . . . 7 ((𝑥 = 𝑎𝑟 = 𝑠) → ((𝐹‘(𝑟 “ {𝑧})) = 𝑧 ↔ (𝐹‘(𝑠 “ {𝑧})) = 𝑧))
202, 19raleqbidv 3364 . . . . . 6 ((𝑥 = 𝑎𝑟 = 𝑠) → (∀𝑧𝑥 (𝐹‘(𝑟 “ {𝑧})) = 𝑧 ↔ ∀𝑧𝑎 (𝐹‘(𝑠 “ {𝑧})) = 𝑧))
2116, 20syl5bb 275 . . . . 5 ((𝑥 = 𝑎𝑟 = 𝑠) → (∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦 ↔ ∀𝑧𝑎 (𝐹‘(𝑠 “ {𝑧})) = 𝑧))
2210, 21anbi12d 624 . . . 4 ((𝑥 = 𝑎𝑟 = 𝑠) → ((𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦) ↔ (𝑠 We 𝑎 ∧ ∀𝑧𝑎 (𝐹‘(𝑠 “ {𝑧})) = 𝑧)))
237, 22anbi12d 624 . . 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 4947 . 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 2849 1 𝑊 = {⟨𝑎, 𝑠⟩ ∣ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧𝑎 (𝐹‘(𝑠 “ {𝑧})) = 𝑧))}
Colors of variables: wff setvar class
Syntax hints:  wa 386   = wceq 1656  wral 3117  wss 3798  {csn 4399  {copab 4937   We wwe 5304   × cxp 5344  ccnv 5345  cima 5349  cfv 6127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-cnv 5354  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fv 6135
This theorem is referenced by:  canthnum  9793  canthp1  9798
  Copyright terms: Public domain W3C validator