Theorem setrec2lem2 47826

Description: Lemma for setrec2 47827. 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 6009	. 2 ⊢ Rel (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
2		fvex 6903	. . . . 5 ⊢ (𝐹‘𝑥) ∈ V
3		eqeq2 2742	. . . . . . 7 ⊢ (𝑧 = (𝐹‘𝑥) → (𝑦 = 𝑧 ↔ 𝑦 = (𝐹‘𝑥)))
4	3	imbi2d 339	. . . . . 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 3595	. . . 4 ⊢ (∀𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 → 𝑦 = (𝐹‘𝑥)) → ∃𝑧∀𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 → 𝑦 = 𝑧))
7		vex 3476	. . . . . 6 ⊢ 𝑦 ∈ V
8	7	brresi 5989	. . . . 5 ⊢ (𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 ↔ (𝑥 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ∧ 𝑥𝐹𝑦))
9		abid 2711	. . . . . . 7 ⊢ (𝑥 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ↔ ∃!𝑦 𝑥𝐹𝑦)
10		tz6.12-1 6913	. . . . . . . 8 ⊢ ((𝑥𝐹𝑦 ∧ ∃!𝑦 𝑥𝐹𝑦) → (𝐹‘𝑥) = 𝑦)
11	10	ancoms 457	. . . . . . 7 ⊢ ((∃!𝑦 𝑥𝐹𝑦 ∧ 𝑥𝐹𝑦) → (𝐹‘𝑥) = 𝑦)
12	9, 11	sylanb 579	. . . . . 6 ⊢ ((𝑥 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ∧ 𝑥𝐹𝑦) → (𝐹‘𝑥) = 𝑦)
13	12	eqcomd 2736	. . . . 5 ⊢ ((𝑥 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ∧ 𝑥𝐹𝑦) → 𝑦 = (𝐹‘𝑥))
14	8, 13	sylbi 216	. . . 4 ⊢ (𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 → 𝑦 = (𝐹‘𝑥))
15	6, 14	mpg 1797	. . 3 ⊢ ∃𝑧∀𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 → 𝑦 = 𝑧)
16	15	ax-gen 1795	. 2 ⊢ ∀𝑥∃𝑧∀𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 → 𝑦 = 𝑧)
17		nfcv 2901	. . . 4 ⊢ Ⅎ𝑥𝐹
18		nfab1 2903	. . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}
19	17, 18	nfres 5982	. . 3 ⊢ Ⅎ𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
20		nfcv 2901	. . . 4 ⊢ Ⅎ𝑦𝐹
21		nfeu1 2580	. . . . 5 ⊢ Ⅎ𝑦∃!𝑦 𝑥𝐹𝑦
22	21	nfab 2907	. . . 4 ⊢ Ⅎ𝑦{𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}
23	20, 22	nfres 5982	. . 3 ⊢ Ⅎ𝑦(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
24		nfcv 2901	. . 3 ⊢ Ⅎ𝑧(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
25	19, 23, 24	dffun3f 47814	. 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 707	1 ⊢ Fun (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})

Syntax hints:   → wi 4   ∧ wa 394  ∀wal 1537   = wceq 1539  ∃wex 1779   ∈ wcel 2104  ∃!weu 2560  {cab 2707   class class class wbr 5147   ↾ cres 5677  Rel wrel 5680  Fun wfun 6536  ‘cfv 6542

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-res 5687  df-iota 6494  df-fun 6544  df-fv 6550

This theorem is referenced by:  setrec2  47827