Theorem acnlem 9999

Description: Construct a mapping satisfying the consequent of isacn 9995. (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 6892	. . . . . 6 ⊢ (𝑓‘𝑥) ⊆ ∪ ran 𝑓
2		simpr 488	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ (𝑓‘𝑥)) → 𝐵 ∈ (𝑓‘𝑥))
3	1, 2	sselid 3934	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ (𝑓‘𝑥)) → 𝐵 ∈ ∪ ran 𝑓)
4	3	ralimiaa 3097	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ (𝑓‘𝑥) → ∀𝑥 ∈ 𝐴 𝐵 ∈ ∪ ran 𝑓)
5		eqid 2761	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
6	5	fmpt 7085	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ ∪ ran 𝑓 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶∪ ran 𝑓)
7	4, 6	sylib 220	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ (𝑓‘𝑥) → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶∪ ran 𝑓)
8		id 22	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉)
9		vex 3457	. . . . . 6 ⊢ 𝑓 ∈ V
10	9	rnex 7885	. . . . 5 ⊢ ran 𝑓 ∈ V
11	10	uniex 7718	. . . 4 ⊢ ∪ ran 𝑓 ∈ V
12		fex2 7911	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶∪ ran 𝑓 ∧ 𝐴 ∈ 𝑉 ∧ ∪ ran 𝑓 ∈ V) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
13	11, 12	mp3an3 1470	. . 3 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶∪ ran 𝑓 ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
14	7, 8, 13	syl2anr 606	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (𝑓‘𝑥)) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
15	5	fvmpt2 6981	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ (𝑓‘𝑥)) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
16	15, 2	eqeltrd 2861	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ (𝑓‘𝑥)) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (𝑓‘𝑥))
17	16	ralimiaa 3097	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ (𝑓‘𝑥) → ∀𝑥 ∈ 𝐴 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (𝑓‘𝑥))
18	17	adantl 485	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (𝑓‘𝑥)) → ∀𝑥 ∈ 𝐴 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (𝑓‘𝑥))
19		nfmpt1 5198	. . . 4 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
20	19	nfeq2 2940	. . 3 ⊢ Ⅎ𝑥 𝑔 = (𝑥 ∈ 𝐴 ↦ 𝐵)
21		fveq1 6860	. . . 4 ⊢ (𝑔 = (𝑥 ∈ 𝐴 ↦ 𝐵) → (𝑔‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
22	21	eleq1d 2846	. . 3 ⊢ (𝑔 = (𝑥 ∈ 𝐴 ↦ 𝐵) → ((𝑔‘𝑥) ∈ (𝑓‘𝑥) ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (𝑓‘𝑥)))
23	20, 22	ralbid 3274	. 2 ⊢ (𝑔 = (𝑥 ∈ 𝐴 ↦ 𝐵) → (∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥) ↔ ∀𝑥 ∈ 𝐴 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ∈ (𝑓‘𝑥)))
24	14, 18, 23	spcedv 3557	1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (𝑓‘𝑥)) → ∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559  ∃wex 1798   ∈ wcel 2141  ∀wral 3075  Vcvv 3453  ∪ cuni 4864   ↦ cmpt 5180  ran crn 5646  ⟶wf 6511  ‘cfv 6515

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-pow 5321  ax-pr 5389  ax-un 7712

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-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-fv 6523

This theorem is referenced by:  numacn  10000  acndom  10002  acndom2  10005