Theorem setrec2lem1 49505

Description: Lemma for setrec2 49507. 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 6894	. 2 ⊢ (𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹‘𝑎))
2		nfvres 6916	. . 3 ⊢ (¬ 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = ∅)
3		vex 3463	. . . . 5 ⊢ 𝑎 ∈ V
4		breq1 5122	. . . . . 6 ⊢ (𝑥 = 𝑎 → (𝑥𝐹𝑦 ↔ 𝑎𝐹𝑦))
5	4	eubidv 2585	. . . . 5 ⊢ (𝑥 = 𝑎 → (∃!𝑦 𝑥𝐹𝑦 ↔ ∃!𝑦 𝑎𝐹𝑦))
6	3, 5	elab 3658	. . . 4 ⊢ (𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ↔ ∃!𝑦 𝑎𝐹𝑦)
7		tz6.12-2 6863	. . . 4 ⊢ (¬ ∃!𝑦 𝑎𝐹𝑦 → (𝐹‘𝑎) = ∅)
8	6, 7	sylnbi 330	. . 3 ⊢ (¬ 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → (𝐹‘𝑎) = ∅)
9	2, 8	eqtr4d 2773	. 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 1540   ∈ wcel 2108  ∃!weu 2567  {cab 2713  ∅c0 4308   class class class wbr 5119   ↾ cres 5656  ‘cfv 6530

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 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-xp 5660  df-dm 5664  df-res 5666  df-iota 6483  df-fv 6538

This theorem is referenced by:  setrec2  49507