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Theorem marypha2lem3 9434
Description: Lemma for marypha2 9436. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)
Hypothesis
Ref Expression
marypha2lem.t 𝑇 = 𝑥𝐴 ({𝑥} × (𝐹𝑥))
Assertion
Ref Expression
marypha2lem3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐺𝑇 ↔ ∀𝑥𝐴 (𝐺𝑥) ∈ (𝐹𝑥)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐺
Allowed substitution hint:   𝑇(𝑥)

Proof of Theorem marypha2lem3
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dffn5 6949 . . . . . . 7 (𝐺 Fn 𝐴𝐺 = (𝑥𝐴 ↦ (𝐺𝑥)))
21biimpi 215 . . . . . 6 (𝐺 Fn 𝐴𝐺 = (𝑥𝐴 ↦ (𝐺𝑥)))
32adantl 480 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → 𝐺 = (𝑥𝐴 ↦ (𝐺𝑥)))
4 df-mpt 5231 . . . . 5 (𝑥𝐴 ↦ (𝐺𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐺𝑥))}
53, 4eqtrdi 2786 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → 𝐺 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐺𝑥))})
6 marypha2lem.t . . . . . 6 𝑇 = 𝑥𝐴 ({𝑥} × (𝐹𝑥))
76marypha2lem2 9433 . . . . 5 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
87a1i 11 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))})
95, 8sseq12d 4014 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐺𝑇 ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐺𝑥))} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}))
10 ssopab2bw 5546 . . 3 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐺𝑥))} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} ↔ ∀𝑥𝑦((𝑥𝐴𝑦 = (𝐺𝑥)) → (𝑥𝐴𝑦 ∈ (𝐹𝑥))))
119, 10bitrdi 286 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐺𝑇 ↔ ∀𝑥𝑦((𝑥𝐴𝑦 = (𝐺𝑥)) → (𝑥𝐴𝑦 ∈ (𝐹𝑥)))))
12 19.21v 1940 . . . . 5 (∀𝑦(𝑥𝐴 → (𝑦 = (𝐺𝑥) → 𝑦 ∈ (𝐹𝑥))) ↔ (𝑥𝐴 → ∀𝑦(𝑦 = (𝐺𝑥) → 𝑦 ∈ (𝐹𝑥))))
13 imdistan 566 . . . . . 6 ((𝑥𝐴 → (𝑦 = (𝐺𝑥) → 𝑦 ∈ (𝐹𝑥))) ↔ ((𝑥𝐴𝑦 = (𝐺𝑥)) → (𝑥𝐴𝑦 ∈ (𝐹𝑥))))
1413albii 1819 . . . . 5 (∀𝑦(𝑥𝐴 → (𝑦 = (𝐺𝑥) → 𝑦 ∈ (𝐹𝑥))) ↔ ∀𝑦((𝑥𝐴𝑦 = (𝐺𝑥)) → (𝑥𝐴𝑦 ∈ (𝐹𝑥))))
15 fvex 6903 . . . . . . 7 (𝐺𝑥) ∈ V
16 eleq1 2819 . . . . . . 7 (𝑦 = (𝐺𝑥) → (𝑦 ∈ (𝐹𝑥) ↔ (𝐺𝑥) ∈ (𝐹𝑥)))
1715, 16ceqsalv 3510 . . . . . 6 (∀𝑦(𝑦 = (𝐺𝑥) → 𝑦 ∈ (𝐹𝑥)) ↔ (𝐺𝑥) ∈ (𝐹𝑥))
1817imbi2i 335 . . . . 5 ((𝑥𝐴 → ∀𝑦(𝑦 = (𝐺𝑥) → 𝑦 ∈ (𝐹𝑥))) ↔ (𝑥𝐴 → (𝐺𝑥) ∈ (𝐹𝑥)))
1912, 14, 183bitr3i 300 . . . 4 (∀𝑦((𝑥𝐴𝑦 = (𝐺𝑥)) → (𝑥𝐴𝑦 ∈ (𝐹𝑥))) ↔ (𝑥𝐴 → (𝐺𝑥) ∈ (𝐹𝑥)))
2019albii 1819 . . 3 (∀𝑥𝑦((𝑥𝐴𝑦 = (𝐺𝑥)) → (𝑥𝐴𝑦 ∈ (𝐹𝑥))) ↔ ∀𝑥(𝑥𝐴 → (𝐺𝑥) ∈ (𝐹𝑥)))
21 df-ral 3060 . . 3 (∀𝑥𝐴 (𝐺𝑥) ∈ (𝐹𝑥) ↔ ∀𝑥(𝑥𝐴 → (𝐺𝑥) ∈ (𝐹𝑥)))
2220, 21bitr4i 277 . 2 (∀𝑥𝑦((𝑥𝐴𝑦 = (𝐺𝑥)) → (𝑥𝐴𝑦 ∈ (𝐹𝑥))) ↔ ∀𝑥𝐴 (𝐺𝑥) ∈ (𝐹𝑥))
2311, 22bitrdi 286 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐺𝑇 ↔ ∀𝑥𝐴 (𝐺𝑥) ∈ (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wal 1537   = wceq 1539  wcel 2104  wral 3059  wss 3947  {csn 4627   ciun 4996  {copab 5209  cmpt 5230   × cxp 5673   Fn wfn 6537  cfv 6542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-fv 6550
This theorem is referenced by:  marypha2  9436
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