Theorem acnlem 10001

Description: Construct a mapping satisfying the consequent of isacn 9997. (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 6891	. . . . . 6 ⊢ (𝑓‘𝑥) ⊆ ∪ ran 𝑓
2		simpr 484	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ (𝑓‘𝑥)) → 𝐵 ∈ (𝑓‘𝑥))
3	1, 2	sselid 3944	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ (𝑓‘𝑥)) → 𝐵 ∈ ∪ ran 𝑓)
4	3	ralimiaa 3065	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ (𝑓‘𝑥) → ∀𝑥 ∈ 𝐴 𝐵 ∈ ∪ ran 𝑓)
5		eqid 2729	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
6	5	fmpt 7082	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ ∪ ran 𝑓 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶∪ ran 𝑓)
7	4, 6	sylib 218	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ (𝑓‘𝑥) → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶∪ ran 𝑓)
8		id 22	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉)
9		vex 3451	. . . . . 6 ⊢ 𝑓 ∈ V
10	9	rnex 7886	. . . . 5 ⊢ ran 𝑓 ∈ V
11	10	uniex 7717	. . . 4 ⊢ ∪ ran 𝑓 ∈ V
12		fex2 7912	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶∪ ran 𝑓 ∧ 𝐴 ∈ 𝑉 ∧ ∪ ran 𝑓 ∈ V) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
13	11, 12	mp3an3 1452	. . 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 6979	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ (𝑓‘𝑥)) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
16	15, 2	eqeltrd 2828	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ (𝑓‘𝑥)) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (𝑓‘𝑥))
17	16	ralimiaa 3065	. . 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 5206	. . . 4 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
20	19	nfeq2 2909	. . 3 ⊢ Ⅎ𝑥 𝑔 = (𝑥 ∈ 𝐴 ↦ 𝐵)
21		fveq1 6857	. . . 4 ⊢ (𝑔 = (𝑥 ∈ 𝐴 ↦ 𝐵) → (𝑔‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
22	21	eleq1d 2813	. . 3 ⊢ (𝑔 = (𝑥 ∈ 𝐴 ↦ 𝐵) → ((𝑔‘𝑥) ∈ (𝑓‘𝑥) ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (𝑓‘𝑥)))
23	20, 22	ralbid 3250	. 2 ⊢ (𝑔 = (𝑥 ∈ 𝐴 ↦ 𝐵) → (∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥) ↔ ∀𝑥 ∈ 𝐴 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (𝑓‘𝑥)))
24	14, 18, 23	spcedv 3564	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (𝑓‘𝑥)) → ∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3044  Vcvv 3447  ∪ cuni 4871   ↦ cmpt 5188  ran crn 5639  ⟶wf 6507  ‘cfv 6511

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519

This theorem is referenced by:  numacn  10002  acndom  10004  acndom2  10007