Theorem setrec2lem2 50455

Description: Lemma for setrec2 50456. 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 6006	. 2 ⊢ Rel (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
2		fvex 6896	. . . . 5 ⊢ (𝐹‘𝑥) ∈ V
3		eqeq2 2775	. . . . . . 7 ⊢ (𝑧 = (𝐹‘𝑥) → (𝑦 = 𝑧 ↔ 𝑦 = (𝐹‘𝑥)))
4	3	imbi2d 343	. . . . . 6 ⊢ (𝑧 = (𝐹‘𝑥) → ((𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 → 𝑦 = 𝑧) ↔ (𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 → 𝑦 = (𝐹‘𝑥))))
5	4	albidv 1950	. . . . 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 3566	. . . 4 ⊢ (∀𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 → 𝑦 = (𝐹‘𝑥)) → ∃𝑧∀𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 → 𝑦 = 𝑧))
7		vex 3459	. . . . . 6 ⊢ 𝑦 ∈ V
8	7	brresi 5989	. . . . 5 ⊢ (𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 ↔ (𝑥 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ∧ 𝑥𝐹𝑦))
9		abid 2745	. . . . . . 7 ⊢ (𝑥 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ↔ ∃!𝑦 𝑥𝐹𝑦)
10		tz6.12-1 6906	. . . . . . . 8 ⊢ ((𝑥𝐹𝑦 ∧ ∃!𝑦 𝑥𝐹𝑦) → (𝐹‘𝑥) = 𝑦)
11	10	ancoms 463	. . . . . . 7 ⊢ ((∃!𝑦 𝑥𝐹𝑦 ∧ 𝑥𝐹𝑦) → (𝐹‘𝑥) = 𝑦)
12	9, 11	sylanb 592	. . . . . 6 ⊢ ((𝑥 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ∧ 𝑥𝐹𝑦) → (𝐹‘𝑥) = 𝑦)
13	12	eqcomd 2769	. . . . 5 ⊢ ((𝑥 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ∧ 𝑥𝐹𝑦) → 𝑦 = (𝐹‘𝑥))
14	8, 13	sylbi 220	. . . 4 ⊢ (𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 → 𝑦 = (𝐹‘𝑥))
15	6, 14	mpg 1827	. . 3 ⊢ ∃𝑧∀𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 → 𝑦 = 𝑧)
16	15	ax-gen 1825	. 2 ⊢ ∀𝑥∃𝑧∀𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 → 𝑦 = 𝑧)
17		nfcv 2925	. . . 4 ⊢ Ⅎ𝑥𝐹
18		nfab1 2927	. . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}
19	17, 18	nfres 5982	. . 3 ⊢ Ⅎ𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
20		nfcv 2925	. . . 4 ⊢ Ⅎ𝑦𝐹
21		nfeu1 2617	. . . . 5 ⊢ Ⅎ𝑦∃!𝑦 𝑥𝐹𝑦
22	21	nfab 2931	. . . 4 ⊢ Ⅎ𝑦{𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}
23	20, 22	nfres 5982	. . 3 ⊢ Ⅎ𝑦(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
24		nfcv 2925	. . 3 ⊢ Ⅎ𝑧(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
25	19, 23, 24	dffun3f 50443	. 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 723	1 ⊢ Fun (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})

Syntax hints:   → wi 4   ∧ wa 400  ∀wal 1568   = wceq 1570  ∃wex 1809   ∈ wcel 2143  ∃!weu 2596  {cab 2741   class class class wbr 5110   ↾ cres 5665  Rel wrel 5668  Fun wfun 6532  ‘cfv 6538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-res 5675  df-iota 6494  df-fun 6540  df-fv 6546

This theorem is referenced by:  setrec2  50456