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Theorem setrec2lem1 49212
Description: Lemma for setrec2 49214. 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 6925 . 2 (𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹𝑎))
2 nfvres 6947 . . 3 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = ∅)
3 vex 3484 . . . . 5 𝑎 ∈ V
4 breq1 5146 . . . . . 6 (𝑥 = 𝑎 → (𝑥𝐹𝑦𝑎𝐹𝑦))
54eubidv 2586 . . . . 5 (𝑥 = 𝑎 → (∃!𝑦 𝑥𝐹𝑦 ↔ ∃!𝑦 𝑎𝐹𝑦))
63, 5elab 3679 . . . 4 (𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ↔ ∃!𝑦 𝑎𝐹𝑦)
7 tz6.12-2 6894 . . . 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 1540  wcel 2108  ∃!weu 2568  {cab 2714  c0 4333   class class class wbr 5143  cres 5687  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-dm 5695  df-res 5697  df-iota 6514  df-fv 6569
This theorem is referenced by:  setrec2  49214
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