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Theorem setrec2lem1 47212
Description: Lemma for setrec2 47214. The functional part of 𝐹 has the same values as 𝐹. (Contributed by Emmett Weisz, 4-Mar-2021.) (New usage is discouraged.)
Assertion
Ref Expression
setrec2lem1 ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹𝑎)
Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝑎,𝑦
Allowed substitution hint:   𝐹(𝑎)

Proof of Theorem setrec2lem1
StepHypRef Expression
1 fvres 6866 . 2 (𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹𝑎))
2 nfvres 6888 . . 3 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = ∅)
3 vex 3452 . . . . 5 𝑎 ∈ V
4 breq1 5113 . . . . . 6 (𝑥 = 𝑎 → (𝑥𝐹𝑦𝑎𝐹𝑦))
54eubidv 2585 . . . . 5 (𝑥 = 𝑎 → (∃!𝑦 𝑥𝐹𝑦 ↔ ∃!𝑦 𝑎𝐹𝑦))
63, 5elab 3635 . . . 4 (𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ↔ ∃!𝑦 𝑎𝐹𝑦)
7 tz6.12-2 6835 . . . 4 (¬ ∃!𝑦 𝑎𝐹𝑦 → (𝐹𝑎) = ∅)
86, 7sylnbi 330 . . 3 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → (𝐹𝑎) = ∅)
92, 8eqtr4d 2780 . 2 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹𝑎))
101, 9pm2.61i 182 1 ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹𝑎)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1542  wcel 2107  ∃!weu 2567  {cab 2714  c0 4287   class class class wbr 5110  cres 5640  cfv 6501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-xp 5644  df-dm 5648  df-res 5650  df-iota 6453  df-fv 6509
This theorem is referenced by:  setrec2  47214
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