Theorem setrec2lem1 48235

Description: Lemma for setrec2 48237. 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) = (𝐹‘𝑎)

Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝑎,𝑦

Allowed substitution hint:   𝐹(𝑎)

Proof of Theorem setrec2lem1

Step	Hyp	Ref	Expression
1		fvres 6910	. 2 ⊢ (𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹‘𝑎))
2		nfvres 6932	. . 3 ⊢ (¬ 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = ∅)
3		vex 3467	. . . . 5 ⊢ 𝑎 ∈ V
4		breq1 5146	. . . . . 6 ⊢ (𝑥 = 𝑎 → (𝑥𝐹𝑦 ↔ 𝑎𝐹𝑦))
5	4	eubidv 2574	. . . . 5 ⊢ (𝑥 = 𝑎 → (∃!𝑦 𝑥𝐹𝑦 ↔ ∃!𝑦 𝑎𝐹𝑦))
6	3, 5	elab 3660	. . . 4 ⊢ (𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ↔ ∃!𝑦 𝑎𝐹𝑦)
7		tz6.12-2 6879	. . . 4 ⊢ (¬ ∃!𝑦 𝑎𝐹𝑦 → (𝐹‘𝑎) = ∅)
8	6, 7	sylnbi 329	. . 3 ⊢ (¬ 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → (𝐹‘𝑎) = ∅)
9	2, 8	eqtr4d 2768	. 2 ⊢ (¬ 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹‘𝑎))
10	1, 9	pm2.61i 182	1 ⊢ ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹‘𝑎)

Syntax hints:  ¬ wn 3   = wceq 1533   ∈ wcel 2098  ∃!weu 2556  {cab 2702  ∅c0 4318   class class class wbr 5143   ↾ cres 5674  ‘cfv 6542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-xp 5678  df-dm 5682  df-res 5684  df-iota 6494  df-fv 6550

This theorem is referenced by:  setrec2  48237