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Theorem marypha2lem3 9373
Description: Lemma for marypha2 9375. 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 6901 . . . . . . 7 (𝐺 Fn 𝐴𝐺 = (𝑥𝐴 ↦ (𝐺𝑥)))
21biimpi 215 . . . . . 6 (𝐺 Fn 𝐴𝐺 = (𝑥𝐴 ↦ (𝐺𝑥)))
32adantl 482 . . . . 5 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → 𝐺 = (𝑥𝐴 ↦ (𝐺𝑥)))
4 df-mpt 5189 . . . . 5 (𝑥𝐴 ↦ (𝐺𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐺𝑥))}
53, 4eqtrdi 2792 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → 𝐺 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐺𝑥))})
6 marypha2lem.t . . . . . 6 𝑇 = 𝑥𝐴 ({𝑥} × (𝐹𝑥))
76marypha2lem2 9372 . . . . 5 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
87a1i 11 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))})
95, 8sseq12d 3977 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐺𝑇 ↔ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐺𝑥))} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}))
10 ssopab2bw 5504 . . 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 1942 . . . . 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 1821 . . . . 5 (∀𝑦(𝑥𝐴 → (𝑦 = (𝐺𝑥) → 𝑦 ∈ (𝐹𝑥))) ↔ ∀𝑦((𝑥𝐴𝑦 = (𝐺𝑥)) → (𝑥𝐴𝑦 ∈ (𝐹𝑥))))
15 fvex 6855 . . . . . . 7 (𝐺𝑥) ∈ V
16 eleq1 2825 . . . . . . 7 (𝑦 = (𝐺𝑥) → (𝑦 ∈ (𝐹𝑥) ↔ (𝐺𝑥) ∈ (𝐹𝑥)))
1715, 16ceqsalv 3481 . . . . . 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 1821 . . 3 (∀𝑥𝑦((𝑥𝐴𝑦 = (𝐺𝑥)) → (𝑥𝐴𝑦 ∈ (𝐹𝑥))) ↔ ∀𝑥(𝑥𝐴 → (𝐺𝑥) ∈ (𝐹𝑥)))
21 df-ral 3065 . . 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 396  wal 1539   = wceq 1541  wcel 2106  wral 3064  wss 3910  {csn 4586   ciun 4954  {copab 5167  cmpt 5188   × cxp 5631   Fn wfn 6491  cfv 6496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-iota 6448  df-fun 6498  df-fn 6499  df-fv 6504
This theorem is referenced by:  marypha2  9375
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