Theorem setrec2lem1 9967

Description: Lemma for setrec2 9970. 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 6902	. 2 ⊢ (𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹‘𝑎))
2		nfvres 6921	. . 3 ⊢ (¬ 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = ∅)
3		vex 3455	. . . . 5 ⊢ 𝑎 ∈ V
4		breq1 5106	. . . . . 6 ⊢ (𝑥 = 𝑎 → (𝑥𝐹𝑦 ↔ 𝑎𝐹𝑦))
5	4	eubidv 2612	. . . . 5 ⊢ (𝑥 = 𝑎 → (∃!𝑦 𝑥𝐹𝑦 ↔ ∃!𝑦 𝑎𝐹𝑦))
6	3, 5	elab 3633	. . . 4 ⊢ (𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ↔ ∃!𝑦 𝑎𝐹𝑦)
7		tz6.12-2 6870	. . . 4 ⊢ (¬ ∃!𝑦 𝑎𝐹𝑦 → (𝐹‘𝑎) = ∅)
8	6, 7	sylnbi 333	. . 3 ⊢ (¬ 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → (𝐹‘𝑎) = ∅)
9	2, 8	eqtr4d 2799	. 2 ⊢ (¬ 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹‘𝑎))
10	1, 9	pm2.61i 184	1 ⊢ ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹‘𝑎)

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2145  ∃!weu 2594  {cab 2739  ∅c0 4279   class class class wbr 5103   ↾ cres 5653  ‘cfv 6537

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-xp 5657  df-dm 5661  df-res 5663  df-iota 6493  df-fv 6545

This theorem is used by:  setrec2  9970