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Theorem marypha2lem3 8889
Description: Lemma for marypha2 8891. 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 6717 . . . . . . 7 (𝐺 Fn 𝐴𝐺 = (𝑥𝐴 ↦ (𝐺𝑥)))
21biimpi 217 . . . . . 6 (𝐺 Fn 𝐴𝐺 = (𝑥𝐴 ↦ (𝐺𝑥)))
32adantl 482 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → 𝐺 = (𝑥𝐴 ↦ (𝐺𝑥)))
4 df-mpt 5138 . . . . 5 (𝑥𝐴 ↦ (𝐺𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐺𝑥))}
53, 4syl6eq 2869 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → 𝐺 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐺𝑥))})
6 marypha2lem.t . . . . . 6 𝑇 = 𝑥𝐴 ({𝑥} × (𝐹𝑥))
76marypha2lem2 8888 . . . . 5 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
87a1i 11 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))})
95, 8sseq12d 3997 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐺𝑇 ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐺𝑥))} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}))
10 ssopab2bw 5425 . . 3 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐺𝑥))} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} ↔ ∀𝑥𝑦((𝑥𝐴𝑦 = (𝐺𝑥)) → (𝑥𝐴𝑦 ∈ (𝐹𝑥))))
119, 10syl6bb 288 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐺𝑇 ↔ ∀𝑥𝑦((𝑥𝐴𝑦 = (𝐺𝑥)) → (𝑥𝐴𝑦 ∈ (𝐹𝑥)))))
12 19.21v 1931 . . . . 5 (∀𝑦(𝑥𝐴 → (𝑦 = (𝐺𝑥) → 𝑦 ∈ (𝐹𝑥))) ↔ (𝑥𝐴 → ∀𝑦(𝑦 = (𝐺𝑥) → 𝑦 ∈ (𝐹𝑥))))
13 imdistan 568 . . . . . 6 ((𝑥𝐴 → (𝑦 = (𝐺𝑥) → 𝑦 ∈ (𝐹𝑥))) ↔ ((𝑥𝐴𝑦 = (𝐺𝑥)) → (𝑥𝐴𝑦 ∈ (𝐹𝑥))))
1413albii 1811 . . . . 5 (∀𝑦(𝑥𝐴 → (𝑦 = (𝐺𝑥) → 𝑦 ∈ (𝐹𝑥))) ↔ ∀𝑦((𝑥𝐴𝑦 = (𝐺𝑥)) → (𝑥𝐴𝑦 ∈ (𝐹𝑥))))
15 fvex 6676 . . . . . . 7 (𝐺𝑥) ∈ V
16 eleq1 2897 . . . . . . 7 (𝑦 = (𝐺𝑥) → (𝑦 ∈ (𝐹𝑥) ↔ (𝐺𝑥) ∈ (𝐹𝑥)))
1715, 16ceqsalv 3530 . . . . . 6 (∀𝑦(𝑦 = (𝐺𝑥) → 𝑦 ∈ (𝐹𝑥)) ↔ (𝐺𝑥) ∈ (𝐹𝑥))
1817imbi2i 337 . . . . 5 ((𝑥𝐴 → ∀𝑦(𝑦 = (𝐺𝑥) → 𝑦 ∈ (𝐹𝑥))) ↔ (𝑥𝐴 → (𝐺𝑥) ∈ (𝐹𝑥)))
1912, 14, 183bitr3i 302 . . . 4 (∀𝑦((𝑥𝐴𝑦 = (𝐺𝑥)) → (𝑥𝐴𝑦 ∈ (𝐹𝑥))) ↔ (𝑥𝐴 → (𝐺𝑥) ∈ (𝐹𝑥)))
2019albii 1811 . . 3 (∀𝑥𝑦((𝑥𝐴𝑦 = (𝐺𝑥)) → (𝑥𝐴𝑦 ∈ (𝐹𝑥))) ↔ ∀𝑥(𝑥𝐴 → (𝐺𝑥) ∈ (𝐹𝑥)))
21 df-ral 3140 . . 3 (∀𝑥𝐴 (𝐺𝑥) ∈ (𝐹𝑥) ↔ ∀𝑥(𝑥𝐴 → (𝐺𝑥) ∈ (𝐹𝑥)))
2220, 21bitr4i 279 . 2 (∀𝑥𝑦((𝑥𝐴𝑦 = (𝐺𝑥)) → (𝑥𝐴𝑦 ∈ (𝐹𝑥))) ↔ ∀𝑥𝐴 (𝐺𝑥) ∈ (𝐹𝑥))
2311, 22syl6bb 288 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐺𝑇 ↔ ∀𝑥𝐴 (𝐺𝑥) ∈ (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1526   = wceq 1528  wcel 2105  wral 3135  wss 3933  {csn 4557   ciun 4910  {copab 5119  cmpt 5137   × cxp 5546   Fn wfn 6343  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fn 6351  df-fv 6356
This theorem is referenced by:  marypha2  8891
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