Theorem acnlem 9804

Description: Construct a mapping satisfying the consequent of isacn 9800. (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 6803	. . . . . 6 ⊢ (𝑓‘𝑥) ⊆ ∪ ran 𝑓
2		simpr 485	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ (𝑓‘𝑥)) → 𝐵 ∈ (𝑓‘𝑥))
3	1, 2	sselid 3919	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ (𝑓‘𝑥)) → 𝐵 ∈ ∪ ran 𝑓)
4	3	ralimiaa 3086	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ (𝑓‘𝑥) → ∀𝑥 ∈ 𝐴 𝐵 ∈ ∪ ran 𝑓)
5		eqid 2738	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
6	5	fmpt 6984	. . . 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 3436	. . . . . 6 ⊢ 𝑓 ∈ V
10	9	rnex 7759	. . . . 5 ⊢ ran 𝑓 ∈ V
11	10	uniex 7594	. . . 4 ⊢ ∪ ran 𝑓 ∈ V
12		fex2 7780	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶∪ ran 𝑓 ∧ 𝐴 ∈ 𝑉 ∧ ∪ ran 𝑓 ∈ V) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
13	11, 12	mp3an3 1449	. . 3 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶∪ ran 𝑓 ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
14	7, 8, 13	syl2anr 597	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (𝑓‘𝑥)) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
15	5	fvmpt2 6886	. . . . 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 3086	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ (𝑓‘𝑥) → ∀𝑥 ∈ 𝐴 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (𝑓‘𝑥))
18	17	adantl 482	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (𝑓‘𝑥)) → ∀𝑥 ∈ 𝐴 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (𝑓‘𝑥))
19		nfmpt1 5182	. . . 4 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
20	19	nfeq2 2924	. . 3 ⊢ Ⅎ𝑥 𝑔 = (𝑥 ∈ 𝐴 ↦ 𝐵)
21		fveq1 6773	. . . 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 3161	. 2 ⊢ (𝑔 = (𝑥 ∈ 𝐴 ↦ 𝐵) → (∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥) ↔ ∀𝑥 ∈ 𝐴 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (𝑓‘𝑥)))
24	14, 18, 23	spcedv 3537	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (𝑓‘𝑥)) → ∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∀wral 3064  Vcvv 3432  ∪ cuni 4839   ↦ cmpt 5157  ran crn 5590  ⟶wf 6429  ‘cfv 6433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441

This theorem is referenced by:  numacn  9805  acndom  9807  acndom2  9810