Theorem setrec2lem2 50191

Description: Lemma for setrec2 50192. The functional part of 𝐹 is a function. (Contributed by Emmett Weisz, 6-Mar-2021.) (New usage is discouraged.)

Assertion

Ref	Expression
setrec2lem2	⊢ Fun (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})

Distinct variable group:   𝑥,𝑦,𝐹

Proof of Theorem setrec2lem2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		relres 5964	. 2 ⊢ Rel (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
2		fvex 6847	. . . . 5 ⊢ (𝐹‘𝑥) ∈ V
3		eqeq2 2752	. . . . . . 7 ⊢ (𝑧 = (𝐹‘𝑥) → (𝑦 = 𝑧 ↔ 𝑦 = (𝐹‘𝑥)))
4	3	imbi2d 341	. . . . . 6 ⊢ (𝑧 = (𝐹‘𝑥) → ((𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 → 𝑦 = 𝑧) ↔ (𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 → 𝑦 = (𝐹‘𝑥))))
5	4	albidv 1927	. . . . 5 ⊢ (𝑧 = (𝐹‘𝑥) → (∀𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 → 𝑦 = 𝑧) ↔ ∀𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 → 𝑦 = (𝐹‘𝑥))))
6	2, 5	spcev 3551	. . . 4 ⊢ (∀𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 → 𝑦 = (𝐹‘𝑥)) → ∃𝑧∀𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 → 𝑦 = 𝑧))
7		vex 3436	. . . . . 6 ⊢ 𝑦 ∈ V
8	7	brresi 5947	. . . . 5 ⊢ (𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 ↔ (𝑥 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ∧ 𝑥𝐹𝑦))
9		abid 2722	. . . . . . 7 ⊢ (𝑥 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ↔ ∃!𝑦 𝑥𝐹𝑦)
10		tz6.12-1 6857	. . . . . . . 8 ⊢ ((𝑥𝐹𝑦 ∧ ∃!𝑦 𝑥𝐹𝑦) → (𝐹‘𝑥) = 𝑦)
11	10	ancoms 459	. . . . . . 7 ⊢ ((∃!𝑦 𝑥𝐹𝑦 ∧ 𝑥𝐹𝑦) → (𝐹‘𝑥) = 𝑦)
12	9, 11	sylanb 587	. . . . . 6 ⊢ ((𝑥 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ∧ 𝑥𝐹𝑦) → (𝐹‘𝑥) = 𝑦)
13	12	eqcomd 2746	. . . . 5 ⊢ ((𝑥 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ∧ 𝑥𝐹𝑦) → 𝑦 = (𝐹‘𝑥))
14	8, 13	sylbi 218	. . . 4 ⊢ (𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 → 𝑦 = (𝐹‘𝑥))
15	6, 14	mpg 1804	. . 3 ⊢ ∃𝑧∀𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 → 𝑦 = 𝑧)
16	15	ax-gen 1802	. 2 ⊢ ∀𝑥∃𝑧∀𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 → 𝑦 = 𝑧)
17		nfcv 2902	. . . 4 ⊢ Ⅎ𝑥𝐹
18		nfab1 2904	. . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}
19	17, 18	nfres 5940	. . 3 ⊢ Ⅎ𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
20		nfcv 2902	. . . 4 ⊢ Ⅎ𝑦𝐹
21		nfeu1 2593	. . . . 5 ⊢ Ⅎ𝑦∃!𝑦 𝑥𝐹𝑦
22	21	nfab 2908	. . . 4 ⊢ Ⅎ𝑦{𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}
23	20, 22	nfres 5940	. . 3 ⊢ Ⅎ𝑦(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
24		nfcv 2902	. . 3 ⊢ Ⅎ𝑧(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
25	19, 23, 24	dffun3f 50179	. 2 ⊢ (Fun (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}) ↔ (Rel (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}) ∧ ∀𝑥∃𝑧∀𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 → 𝑦 = 𝑧)))
26	1, 16, 25	mpbir2an 717	1 ⊢ Fun (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1545   = wceq 1547  ∃wex 1786   ∈ wcel 2119  ∃!weu 2572  {cab 2718   class class class wbr 5079   ↾ cres 5627  Rel wrel 5630  Fun wfun 6486  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-res 5637  df-iota 6448  df-fun 6494  df-fv 6500

This theorem is referenced by:  setrec2  50192