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Theorem setrec2lem2 47129
Description: Lemma for setrec2 47130. The functional part of 𝐹 is a function. (Contributed by Emmett Weisz, 6-Mar-2021.) (New usage is discouraged.)
Assertion
Ref Expression
setrec2lem2 Fun (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
Distinct variable group:   𝑥,𝑦,𝐹

Proof of Theorem setrec2lem2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 relres 5966 . 2 Rel (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
2 fvex 6855 . . . . 5 (𝐹𝑥) ∈ V
3 eqeq2 2748 . . . . . . 7 (𝑧 = (𝐹𝑥) → (𝑦 = 𝑧𝑦 = (𝐹𝑥)))
43imbi2d 340 . . . . . 6 (𝑧 = (𝐹𝑥) → ((𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = 𝑧) ↔ (𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = (𝐹𝑥))))
54albidv 1923 . . . . 5 (𝑧 = (𝐹𝑥) → (∀𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = 𝑧) ↔ ∀𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = (𝐹𝑥))))
62, 5spcev 3565 . . . 4 (∀𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = (𝐹𝑥)) → ∃𝑧𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = 𝑧))
7 vex 3449 . . . . . 6 𝑦 ∈ V
87brresi 5946 . . . . 5 (𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 ↔ (𝑥 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ∧ 𝑥𝐹𝑦))
9 abid 2717 . . . . . . 7 (𝑥 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ↔ ∃!𝑦 𝑥𝐹𝑦)
10 tz6.12-1 6865 . . . . . . . 8 ((𝑥𝐹𝑦 ∧ ∃!𝑦 𝑥𝐹𝑦) → (𝐹𝑥) = 𝑦)
1110ancoms 459 . . . . . . 7 ((∃!𝑦 𝑥𝐹𝑦𝑥𝐹𝑦) → (𝐹𝑥) = 𝑦)
129, 11sylanb 581 . . . . . 6 ((𝑥 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ∧ 𝑥𝐹𝑦) → (𝐹𝑥) = 𝑦)
1312eqcomd 2742 . . . . 5 ((𝑥 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ∧ 𝑥𝐹𝑦) → 𝑦 = (𝐹𝑥))
148, 13sylbi 216 . . . 4 (𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = (𝐹𝑥))
156, 14mpg 1799 . . 3 𝑧𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = 𝑧)
1615ax-gen 1797 . 2 𝑥𝑧𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = 𝑧)
17 nfcv 2907 . . . 4 𝑥𝐹
18 nfab1 2909 . . . 4 𝑥{𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}
1917, 18nfres 5939 . . 3 𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
20 nfcv 2907 . . . 4 𝑦𝐹
21 nfeu1 2586 . . . . 5 𝑦∃!𝑦 𝑥𝐹𝑦
2221nfab 2913 . . . 4 𝑦{𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}
2320, 22nfres 5939 . . 3 𝑦(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
24 nfcv 2907 . . 3 𝑧(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
2519, 23, 24dffun3f 47117 . 2 (Fun (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}) ↔ (Rel (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}) ∧ ∀𝑥𝑧𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = 𝑧)))
261, 16, 25mpbir2an 709 1 Fun (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1539   = wceq 1541  wex 1781  wcel 2106  ∃!weu 2566  {cab 2713   class class class wbr 5105  cres 5635  Rel wrel 5638  Fun wfun 6490  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-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-res 5645  df-iota 6448  df-fun 6498  df-fv 6504
This theorem is referenced by:  setrec2  47130
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