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Theorem marypha2lem3 8384
Description: Lemma for marypha2 8386. 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 6280 . . . . . . 7 (𝐺 Fn 𝐴𝐺 = (𝑥𝐴 ↦ (𝐺𝑥)))
21biimpi 206 . . . . . 6 (𝐺 Fn 𝐴𝐺 = (𝑥𝐴 ↦ (𝐺𝑥)))
32adantl 481 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → 𝐺 = (𝑥𝐴 ↦ (𝐺𝑥)))
4 df-mpt 4763 . . . . 5 (𝑥𝐴 ↦ (𝐺𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐺𝑥))}
53, 4syl6eq 2701 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → 𝐺 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐺𝑥))})
6 marypha2lem.t . . . . . 6 𝑇 = 𝑥𝐴 ({𝑥} × (𝐹𝑥))
76marypha2lem2 8383 . . . . 5 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
87a1i 11 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))})
95, 8sseq12d 3667 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐺𝑇 ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐺𝑥))} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}))
10 ssopab2b 5031 . . 3 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐺𝑥))} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} ↔ ∀𝑥𝑦((𝑥𝐴𝑦 = (𝐺𝑥)) → (𝑥𝐴𝑦 ∈ (𝐹𝑥))))
119, 10syl6bb 276 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐺𝑇 ↔ ∀𝑥𝑦((𝑥𝐴𝑦 = (𝐺𝑥)) → (𝑥𝐴𝑦 ∈ (𝐹𝑥)))))
12 19.21v 1908 . . . . 5 (∀𝑦(𝑥𝐴 → (𝑦 = (𝐺𝑥) → 𝑦 ∈ (𝐹𝑥))) ↔ (𝑥𝐴 → ∀𝑦(𝑦 = (𝐺𝑥) → 𝑦 ∈ (𝐹𝑥))))
13 imdistan 725 . . . . . 6 ((𝑥𝐴 → (𝑦 = (𝐺𝑥) → 𝑦 ∈ (𝐹𝑥))) ↔ ((𝑥𝐴𝑦 = (𝐺𝑥)) → (𝑥𝐴𝑦 ∈ (𝐹𝑥))))
1413albii 1787 . . . . 5 (∀𝑦(𝑥𝐴 → (𝑦 = (𝐺𝑥) → 𝑦 ∈ (𝐹𝑥))) ↔ ∀𝑦((𝑥𝐴𝑦 = (𝐺𝑥)) → (𝑥𝐴𝑦 ∈ (𝐹𝑥))))
15 fvex 6239 . . . . . . 7 (𝐺𝑥) ∈ V
16 eleq1 2718 . . . . . . 7 (𝑦 = (𝐺𝑥) → (𝑦 ∈ (𝐹𝑥) ↔ (𝐺𝑥) ∈ (𝐹𝑥)))
1715, 16ceqsalv 3264 . . . . . 6 (∀𝑦(𝑦 = (𝐺𝑥) → 𝑦 ∈ (𝐹𝑥)) ↔ (𝐺𝑥) ∈ (𝐹𝑥))
1817imbi2i 325 . . . . 5 ((𝑥𝐴 → ∀𝑦(𝑦 = (𝐺𝑥) → 𝑦 ∈ (𝐹𝑥))) ↔ (𝑥𝐴 → (𝐺𝑥) ∈ (𝐹𝑥)))
1912, 14, 183bitr3i 290 . . . 4 (∀𝑦((𝑥𝐴𝑦 = (𝐺𝑥)) → (𝑥𝐴𝑦 ∈ (𝐹𝑥))) ↔ (𝑥𝐴 → (𝐺𝑥) ∈ (𝐹𝑥)))
2019albii 1787 . . 3 (∀𝑥𝑦((𝑥𝐴𝑦 = (𝐺𝑥)) → (𝑥𝐴𝑦 ∈ (𝐹𝑥))) ↔ ∀𝑥(𝑥𝐴 → (𝐺𝑥) ∈ (𝐹𝑥)))
21 df-ral 2946 . . 3 (∀𝑥𝐴 (𝐺𝑥) ∈ (𝐹𝑥) ↔ ∀𝑥(𝑥𝐴 → (𝐺𝑥) ∈ (𝐹𝑥)))
2220, 21bitr4i 267 . 2 (∀𝑥𝑦((𝑥𝐴𝑦 = (𝐺𝑥)) → (𝑥𝐴𝑦 ∈ (𝐹𝑥))) ↔ ∀𝑥𝐴 (𝐺𝑥) ∈ (𝐹𝑥))
2311, 22syl6bb 276 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐺𝑇 ↔ ∀𝑥𝐴 (𝐺𝑥) ∈ (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1521   = wceq 1523  wcel 2030  wral 2941  wss 3607  {csn 4210   ciun 4552  {copab 4745  cmpt 4762   × cxp 5141   Fn wfn 5921  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934
This theorem is referenced by:  marypha2  8386
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