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Theorem setrec2lem2 48440
Description: Lemma for setrec2 48441. 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 6015 . 2 Rel (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
2 fvex 6914 . . . . 5 (𝐹𝑥) ∈ V
3 eqeq2 2738 . . . . . . 7 (𝑧 = (𝐹𝑥) → (𝑦 = 𝑧𝑦 = (𝐹𝑥)))
43imbi2d 339 . . . . . 6 (𝑧 = (𝐹𝑥) → ((𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = 𝑧) ↔ (𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = (𝐹𝑥))))
54albidv 1916 . . . . 5 (𝑧 = (𝐹𝑥) → (∀𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = 𝑧) ↔ ∀𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = (𝐹𝑥))))
62, 5spcev 3592 . . . 4 (∀𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = (𝐹𝑥)) → ∃𝑧𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = 𝑧))
7 vex 3466 . . . . . 6 𝑦 ∈ V
87brresi 5998 . . . . 5 (𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 ↔ (𝑥 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ∧ 𝑥𝐹𝑦))
9 abid 2707 . . . . . . 7 (𝑥 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ↔ ∃!𝑦 𝑥𝐹𝑦)
10 tz6.12-1 6924 . . . . . . . 8 ((𝑥𝐹𝑦 ∧ ∃!𝑦 𝑥𝐹𝑦) → (𝐹𝑥) = 𝑦)
1110ancoms 457 . . . . . . 7 ((∃!𝑦 𝑥𝐹𝑦𝑥𝐹𝑦) → (𝐹𝑥) = 𝑦)
129, 11sylanb 579 . . . . . 6 ((𝑥 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ∧ 𝑥𝐹𝑦) → (𝐹𝑥) = 𝑦)
1312eqcomd 2732 . . . . 5 ((𝑥 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ∧ 𝑥𝐹𝑦) → 𝑦 = (𝐹𝑥))
148, 13sylbi 216 . . . 4 (𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = (𝐹𝑥))
156, 14mpg 1792 . . 3 𝑧𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = 𝑧)
1615ax-gen 1790 . 2 𝑥𝑧𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = 𝑧)
17 nfcv 2892 . . . 4 𝑥𝐹
18 nfab1 2894 . . . 4 𝑥{𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}
1917, 18nfres 5991 . . 3 𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
20 nfcv 2892 . . . 4 𝑦𝐹
21 nfeu1 2577 . . . . 5 𝑦∃!𝑦 𝑥𝐹𝑦
2221nfab 2898 . . . 4 𝑦{𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}
2320, 22nfres 5991 . . 3 𝑦(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
24 nfcv 2892 . . 3 𝑧(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
2519, 23, 24dffun3f 48428 . 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 394  wal 1532   = wceq 1534  wex 1774  wcel 2099  ∃!weu 2557  {cab 2703   class class class wbr 5153  cres 5684  Rel wrel 5687  Fun wfun 6548  cfv 6554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-res 5694  df-iota 6506  df-fun 6556  df-fv 6562
This theorem is referenced by:  setrec2  48441
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