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Theorem marypha2lem3 8499
Description: Lemma for marypha2 8501. 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 6383 . . . . . . 7 (𝐺 Fn 𝐴𝐺 = (𝑥𝐴 ↦ (𝐺𝑥)))
21biimpi 206 . . . . . 6 (𝐺 Fn 𝐴𝐺 = (𝑥𝐴 ↦ (𝐺𝑥)))
32adantl 467 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → 𝐺 = (𝑥𝐴 ↦ (𝐺𝑥)))
4 df-mpt 4864 . . . . 5 (𝑥𝐴 ↦ (𝐺𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐺𝑥))}
53, 4syl6eq 2821 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → 𝐺 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐺𝑥))})
6 marypha2lem.t . . . . . 6 𝑇 = 𝑥𝐴 ({𝑥} × (𝐹𝑥))
76marypha2lem2 8498 . . . . 5 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
87a1i 11 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))})
95, 8sseq12d 3783 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐺𝑇 ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐺𝑥))} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}))
10 ssopab2b 5135 . . 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 2020 . . . . 5 (∀𝑦(𝑥𝐴 → (𝑦 = (𝐺𝑥) → 𝑦 ∈ (𝐹𝑥))) ↔ (𝑥𝐴 → ∀𝑦(𝑦 = (𝐺𝑥) → 𝑦 ∈ (𝐹𝑥))))
13 imdistan 557 . . . . . 6 ((𝑥𝐴 → (𝑦 = (𝐺𝑥) → 𝑦 ∈ (𝐹𝑥))) ↔ ((𝑥𝐴𝑦 = (𝐺𝑥)) → (𝑥𝐴𝑦 ∈ (𝐹𝑥))))
1413albii 1895 . . . . 5 (∀𝑦(𝑥𝐴 → (𝑦 = (𝐺𝑥) → 𝑦 ∈ (𝐹𝑥))) ↔ ∀𝑦((𝑥𝐴𝑦 = (𝐺𝑥)) → (𝑥𝐴𝑦 ∈ (𝐹𝑥))))
15 fvex 6342 . . . . . . 7 (𝐺𝑥) ∈ V
16 eleq1 2838 . . . . . . 7 (𝑦 = (𝐺𝑥) → (𝑦 ∈ (𝐹𝑥) ↔ (𝐺𝑥) ∈ (𝐹𝑥)))
1715, 16ceqsalv 3385 . . . . . 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 1895 . . 3 (∀𝑥𝑦((𝑥𝐴𝑦 = (𝐺𝑥)) → (𝑥𝐴𝑦 ∈ (𝐹𝑥))) ↔ ∀𝑥(𝑥𝐴 → (𝐺𝑥) ∈ (𝐹𝑥)))
21 df-ral 3066 . . 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 382  wal 1629   = wceq 1631  wcel 2145  wral 3061  wss 3723  {csn 4316   ciun 4654  {copab 4846  cmpt 4863   × cxp 5247   Fn wfn 6026  cfv 6031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fn 6034  df-fv 6039
This theorem is referenced by:  marypha2  8501
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