Theorem setrec2lem2 50279

Description: Lemma for setrec2 50280. 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 5989	. 2 ⊢ Rel (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
2		fvex 6876	. . . . 5 ⊢ (𝐹‘𝑥) ∈ V
3		eqeq2 2773	. . . . . . 7 ⊢ (𝑧 = (𝐹‘𝑥) → (𝑦 = 𝑧 ↔ 𝑦 = (𝐹‘𝑥)))
4	3	imbi2d 342	. . . . . 6 ⊢ (𝑧 = (𝐹‘𝑥) → ((𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 → 𝑦 = 𝑧) ↔ (𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 → 𝑦 = (𝐹‘𝑥))))
5	4	albidv 1939	. . . . 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 3565	. . . 4 ⊢ (∀𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 → 𝑦 = (𝐹‘𝑥)) → ∃𝑧∀𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 → 𝑦 = 𝑧))
7		vex 3457	. . . . . 6 ⊢ 𝑦 ∈ V
8	7	brresi 5972	. . . . 5 ⊢ (𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 ↔ (𝑥 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ∧ 𝑥𝐹𝑦))
9		abid 2743	. . . . . . 7 ⊢ (𝑥 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ↔ ∃!𝑦 𝑥𝐹𝑦)
10		tz6.12-1 6886	. . . . . . . 8 ⊢ ((𝑥𝐹𝑦 ∧ ∃!𝑦 𝑥𝐹𝑦) → (𝐹‘𝑥) = 𝑦)
11	10	ancoms 462	. . . . . . 7 ⊢ ((∃!𝑦 𝑥𝐹𝑦 ∧ 𝑥𝐹𝑦) → (𝐹‘𝑥) = 𝑦)
12	9, 11	sylanb 590	. . . . . 6 ⊢ ((𝑥 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ∧ 𝑥𝐹𝑦) → (𝐹‘𝑥) = 𝑦)
13	12	eqcomd 2767	. . . . 5 ⊢ ((𝑥 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ∧ 𝑥𝐹𝑦) → 𝑦 = (𝐹‘𝑥))
14	8, 13	sylbi 219	. . . 4 ⊢ (𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 → 𝑦 = (𝐹‘𝑥))
15	6, 14	mpg 1816	. . 3 ⊢ ∃𝑧∀𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 → 𝑦 = 𝑧)
16	15	ax-gen 1814	. 2 ⊢ ∀𝑥∃𝑧∀𝑦(𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})𝑦 → 𝑦 = 𝑧)
17		nfcv 2923	. . . 4 ⊢ Ⅎ𝑥𝐹
18		nfab1 2925	. . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}
19	17, 18	nfres 5965	. . 3 ⊢ Ⅎ𝑥(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
20		nfcv 2923	. . . 4 ⊢ Ⅎ𝑦𝐹
21		nfeu1 2615	. . . . 5 ⊢ Ⅎ𝑦∃!𝑦 𝑥𝐹𝑦
22	21	nfab 2929	. . . 4 ⊢ Ⅎ𝑦{𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}
23	20, 22	nfres 5965	. . 3 ⊢ Ⅎ𝑦(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
24		nfcv 2923	. . 3 ⊢ Ⅎ𝑧(𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})
25	19, 23, 24	dffun3f 50267	. 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 721	1 ⊢ Fun (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})

Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1557   = wceq 1559  ∃wex 1798   ∈ wcel 2141  ∃!weu 2594  {cab 2739   class class class wbr 5099   ↾ cres 5647  Rel wrel 5650  Fun wfun 6511  ‘cfv 6517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-res 5657  df-iota 6473  df-fun 6519  df-fv 6525

This theorem is referenced by:  setrec2  50280