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Theorem setrec2lem2 45150
Description: Lemma for setrec2 45151. 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 5869 . 2 Rel (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
2 fvex 6674 . . . . 5 (𝐹𝑥) ∈ V
3 eqeq2 2836 . . . . . . 7 (𝑧 = (𝐹𝑥) → (𝑦 = 𝑧𝑦 = (𝐹𝑥)))
43imbi2d 344 . . . . . 6 (𝑧 = (𝐹𝑥) → ((𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = 𝑧) ↔ (𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = (𝐹𝑥))))
54albidv 1922 . . . . 5 (𝑧 = (𝐹𝑥) → (∀𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = 𝑧) ↔ ∀𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = (𝐹𝑥))))
62, 5spcev 3593 . . . 4 (∀𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = (𝐹𝑥)) → ∃𝑧𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = 𝑧))
7 vex 3483 . . . . . 6 𝑦 ∈ V
87brresi 5849 . . . . 5 (𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 ↔ (𝑥 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ∧ 𝑥𝐹𝑦))
9 abid 2806 . . . . . . 7 (𝑥 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ↔ ∃!𝑦 𝑥𝐹𝑦)
10 tz6.12-1 6683 . . . . . . . 8 ((𝑥𝐹𝑦 ∧ ∃!𝑦 𝑥𝐹𝑦) → (𝐹𝑥) = 𝑦)
1110ancoms 462 . . . . . . 7 ((∃!𝑦 𝑥𝐹𝑦𝑥𝐹𝑦) → (𝐹𝑥) = 𝑦)
129, 11sylanb 584 . . . . . 6 ((𝑥 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ∧ 𝑥𝐹𝑦) → (𝐹𝑥) = 𝑦)
1312eqcomd 2830 . . . . 5 ((𝑥 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ∧ 𝑥𝐹𝑦) → 𝑦 = (𝐹𝑥))
148, 13sylbi 220 . . . 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 2982 . . . 4 𝑥𝐹
18 nfab1 2984 . . . 4 𝑥{𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}
1917, 18nfres 5842 . . 3 𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
20 nfcv 2982 . . . 4 𝑦𝐹
21 nfeu1 2675 . . . . 5 𝑦∃!𝑦 𝑥𝐹𝑦
2221nfab 2988 . . . 4 𝑦{𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}
2320, 22nfres 5842 . . 3 𝑦(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
24 nfcv 2982 . . 3 𝑧(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
2519, 23, 24dffun3f 45138 . 2 (Fun (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}) ↔ (Rel (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}) ∧ ∀𝑥𝑧𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦𝑦 = 𝑧)))
261, 16, 25mpbir2an 710 1 Fun (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wal 1536   = wceq 1538  wex 1781  wcel 2115  ∃!weu 2654  {cab 2802   class class class wbr 5052  cres 5544  Rel wrel 5547  Fun wfun 6337  cfv 6343
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-res 5554  df-iota 6302  df-fun 6345  df-fv 6351
This theorem is referenced by:  setrec2  45151
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