Theorem setrec2lem2 49725

Description: Lemma for setrec2 49726. 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 5954	. 2 ⊢ Rel (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
2		fvex 6835	. . . . 5 ⊢ (𝐹‘𝑥) ∈ V
3		eqeq2 2743	. . . . . . 7 ⊢ (𝑧 = (𝐹‘𝑥) → (𝑦 = 𝑧 ↔ 𝑦 = (𝐹‘𝑥)))
4	3	imbi2d 340	. . . . . 6 ⊢ (𝑧 = (𝐹‘𝑥) → ((𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 → 𝑦 = 𝑧) ↔ (𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 → 𝑦 = (𝐹‘𝑥))))
5	4	albidv 1921	. . . . 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 3561	. . . 4 ⊢ (∀𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 → 𝑦 = (𝐹‘𝑥)) → ∃𝑧∀𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 → 𝑦 = 𝑧))
7		vex 3440	. . . . . 6 ⊢ 𝑦 ∈ V
8	7	brresi 5937	. . . . 5 ⊢ (𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 ↔ (𝑥 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ∧ 𝑥𝐹𝑦))
9		abid 2713	. . . . . . 7 ⊢ (𝑥 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ↔ ∃!𝑦 𝑥𝐹𝑦)
10		tz6.12-1 6845	. . . . . . . 8 ⊢ ((𝑥𝐹𝑦 ∧ ∃!𝑦 𝑥𝐹𝑦) → (𝐹‘𝑥) = 𝑦)
11	10	ancoms 458	. . . . . . 7 ⊢ ((∃!𝑦 𝑥𝐹𝑦 ∧ 𝑥𝐹𝑦) → (𝐹‘𝑥) = 𝑦)
12	9, 11	sylanb 581	. . . . . 6 ⊢ ((𝑥 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ∧ 𝑥𝐹𝑦) → (𝐹‘𝑥) = 𝑦)
13	12	eqcomd 2737	. . . . 5 ⊢ ((𝑥 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ∧ 𝑥𝐹𝑦) → 𝑦 = (𝐹‘𝑥))
14	8, 13	sylbi 217	. . . 4 ⊢ (𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 → 𝑦 = (𝐹‘𝑥))
15	6, 14	mpg 1798	. . 3 ⊢ ∃𝑧∀𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 → 𝑦 = 𝑧)
16	15	ax-gen 1796	. 2 ⊢ ∀𝑥∃𝑧∀𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 → 𝑦 = 𝑧)
17		nfcv 2894	. . . 4 ⊢ Ⅎ𝑥𝐹
18		nfab1 2896	. . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}
19	17, 18	nfres 5930	. . 3 ⊢ Ⅎ𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
20		nfcv 2894	. . . 4 ⊢ Ⅎ𝑦𝐹
21		nfeu1 2583	. . . . 5 ⊢ Ⅎ𝑦∃!𝑦 𝑥𝐹𝑦
22	21	nfab 2900	. . . 4 ⊢ Ⅎ𝑦{𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}
23	20, 22	nfres 5930	. . 3 ⊢ Ⅎ𝑦(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
24		nfcv 2894	. . 3 ⊢ Ⅎ𝑧(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
25	19, 23, 24	dffun3f 49713	. 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 711	1 ⊢ Fun (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2111  ∃!weu 2563  {cab 2709   class class class wbr 5091   ↾ cres 5618  Rel wrel 5621  Fun wfun 6475  ‘cfv 6481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-res 5628  df-iota 6437  df-fun 6483  df-fv 6489

This theorem is referenced by:  setrec2  49726