Theorem setrec2lem1 43335

Description: Lemma for setrec2 43337. 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 6452	. 2 ⊢ (𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹‘𝑎))
2		dmres 5655	. . . . . . 7 ⊢ dom (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}) = ({𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ∩ dom 𝐹)
3		inss1 4057	. . . . . . 7 ⊢ ({𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ∩ dom 𝐹) ⊆ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}
4	2, 3	eqsstri 3860	. . . . . 6 ⊢ dom (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}) ⊆ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}
5	4	sseli 3823	. . . . 5 ⊢ (𝑎 ∈ dom (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}) → 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
6	5	con3i 152	. . . 4 ⊢ (¬ 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → ¬ 𝑎 ∈ dom (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}))
7		ndmfv 6463	. . . 4 ⊢ (¬ 𝑎 ∈ dom (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}) → ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = ∅)
8	6, 7	syl 17	. . 3 ⊢ (¬ 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = ∅)
9		vex 3417	. . . . . . 7 ⊢ 𝑎 ∈ V
10		breq1 4876	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → (𝑥𝐹𝑦 ↔ 𝑎𝐹𝑦))
11	10	eubidv 2659	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (∃!𝑦 𝑥𝐹𝑦 ↔ ∃!𝑦 𝑎𝐹𝑦))
12	9, 11	elab 3571	. . . . . 6 ⊢ (𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ↔ ∃!𝑦 𝑎𝐹𝑦)
13	12	notbii 312	. . . . 5 ⊢ (¬ 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ↔ ¬ ∃!𝑦 𝑎𝐹𝑦)
14	13	biimpi 208	. . . 4 ⊢ (¬ 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → ¬ ∃!𝑦 𝑎𝐹𝑦)
15		tz6.12-2 6423	. . . 4 ⊢ (¬ ∃!𝑦 𝑎𝐹𝑦 → (𝐹‘𝑎) = ∅)
16	14, 15	syl 17	. . 3 ⊢ (¬ 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → (𝐹‘𝑎) = ∅)
17	8, 16	eqtr4d 2864	. 2 ⊢ (¬ 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹‘𝑎))
18	1, 17	pm2.61i 177	1 ⊢ ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹‘𝑎)

Syntax hints:  ¬ wn 3   = wceq 1658   ∈ wcel 2166  ∃!weu 2639  {cab 2811   ∩ cin 3797  ∅c0 4144   class class class wbr 4873  dom cdm 5342   ↾ cres 5344  ‘cfv 6123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-xp 5348  df-dm 5352  df-res 5354  df-iota 6086  df-fv 6131

This theorem is referenced by:  setrec2  43337