Theorem acnlem 9735

Description: Construct a mapping satisfying the consequent of isacn 9731. (Contributed by Mario Carneiro, 31-Aug-2015.)

Assertion

Ref	Expression
acnlem	⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (𝑓‘𝑥)) → ∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥))

Distinct variable groups:   𝑓,𝑔,𝑥,𝐴   𝐵,𝑔

Allowed substitution hints:   𝐵(𝑥,𝑓)   𝑉(𝑥,𝑓,𝑔)

Proof of Theorem acnlem

Step	Hyp	Ref	Expression
1		fvssunirn 6785	. . . . . 6 ⊢ (𝑓‘𝑥) ⊆ ∪ ran 𝑓
2		simpr 484	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ (𝑓‘𝑥)) → 𝐵 ∈ (𝑓‘𝑥))
3	1, 2	sselid 3915	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ (𝑓‘𝑥)) → 𝐵 ∈ ∪ ran 𝑓)
4	3	ralimiaa 3085	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ (𝑓‘𝑥) → ∀𝑥 ∈ 𝐴 𝐵 ∈ ∪ ran 𝑓)
5		eqid 2738	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
6	5	fmpt 6966	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ ∪ ran 𝑓 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶∪ ran 𝑓)
7	4, 6	sylib 217	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ (𝑓‘𝑥) → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶∪ ran 𝑓)
8		id 22	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉)
9		vex 3426	. . . . . 6 ⊢ 𝑓 ∈ V
10	9	rnex 7733	. . . . 5 ⊢ ran 𝑓 ∈ V
11	10	uniex 7572	. . . 4 ⊢ ∪ ran 𝑓 ∈ V
12		fex2 7754	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶∪ ran 𝑓 ∧ 𝐴 ∈ 𝑉 ∧ ∪ ran 𝑓 ∈ V) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
13	11, 12	mp3an3 1448	. . 3 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶∪ ran 𝑓 ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
14	7, 8, 13	syl2anr 596	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (𝑓‘𝑥)) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
15	5	fvmpt2 6868	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ (𝑓‘𝑥)) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
16	15, 2	eqeltrd 2839	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ (𝑓‘𝑥)) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (𝑓‘𝑥))
17	16	ralimiaa 3085	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ (𝑓‘𝑥) → ∀𝑥 ∈ 𝐴 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (𝑓‘𝑥))
18	17	adantl 481	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (𝑓‘𝑥)) → ∀𝑥 ∈ 𝐴 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (𝑓‘𝑥))
19		nfmpt1 5178	. . . 4 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
20	19	nfeq2 2923	. . 3 ⊢ Ⅎ𝑥 𝑔 = (𝑥 ∈ 𝐴 ↦ 𝐵)
21		fveq1 6755	. . . 4 ⊢ (𝑔 = (𝑥 ∈ 𝐴 ↦ 𝐵) → (𝑔‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
22	21	eleq1d 2823	. . 3 ⊢ (𝑔 = (𝑥 ∈ 𝐴 ↦ 𝐵) → ((𝑔‘𝑥) ∈ (𝑓‘𝑥) ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (𝑓‘𝑥)))
23	20, 22	ralbid 3158	. 2 ⊢ (𝑔 = (𝑥 ∈ 𝐴 ↦ 𝐵) → (∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥) ↔ ∀𝑥 ∈ 𝐴 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (𝑓‘𝑥)))
24	14, 18, 23	spcedv 3527	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (𝑓‘𝑥)) → ∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  ∀wral 3063  Vcvv 3422  ∪ cuni 4836   ↦ cmpt 5153  ran crn 5581  ⟶wf 6414  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426

This theorem is referenced by:  numacn  9736  acndom  9738  acndom2  9741