Theorem acunirnmpt2f 32688

Description: Axiom of choice for the union of the range of a mapping to function. (Contributed by Thierry Arnoux, 7-Nov-2019.)

Hypotheses

Ref	Expression
acunirnmpt.0	⊢ (𝜑 → 𝐴 ∈ 𝑉)
acunirnmpt.1	⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ≠ ∅)
aciunf1lem.a	⊢ Ⅎ𝑗𝐴
acunirnmpt2f.c	⊢ Ⅎ𝑗𝐶
acunirnmpt2f.d	⊢ Ⅎ𝑗𝐷
acunirnmpt2f.2	⊢ 𝐶 = ∪ 𝑗 ∈ 𝐴 𝐵
acunirnmpt2f.3	⊢ (𝑗 = (𝑓‘𝑥) → 𝐵 = 𝐷)
acunirnmpt2f.4	⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ 𝑊)

Assertion

Ref	Expression
acunirnmpt2f	⊢ (𝜑 → ∃𝑓(𝑓:𝐶⟶𝐴 ∧ ∀𝑥 ∈ 𝐶 𝑥 ∈ 𝐷))

Distinct variable groups:   𝑥,𝑓,𝐴   𝐵,𝑓   𝐶,𝑓,𝑥   𝑓,𝑗,𝜑,𝑥

Allowed substitution hints:   𝐴(𝑗)   𝐵(𝑥,𝑗)   𝐶(𝑗)   𝐷(𝑥,𝑓,𝑗)   𝑉(𝑥,𝑓,𝑗)   𝑊(𝑥,𝑓,𝑗)

Proof of Theorem acunirnmpt2f

Dummy variables 𝑐 𝑦 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 768	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑥 ∈ 𝑦) → 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵))
2		vex 3442	. . . . . . 7 ⊢ 𝑦 ∈ V
3		eqid 2734	. . . . . . . 8 ⊢ (𝑗 ∈ 𝐴 ↦ 𝐵) = (𝑗 ∈ 𝐴 ↦ 𝐵)
4	3	elrnmpt 5905	. . . . . . 7 ⊢ (𝑦 ∈ V → (𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑗 ∈ 𝐴 𝑦 = 𝐵))
5	2, 4	ax-mp 5	. . . . . 6 ⊢ (𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑗 ∈ 𝐴 𝑦 = 𝐵)
6	1, 5	sylib 218	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑥 ∈ 𝑦) → ∃𝑗 ∈ 𝐴 𝑦 = 𝐵)
7		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑗𝜑
8		acunirnmpt2f.c	. . . . . . . . . 10 ⊢ Ⅎ𝑗𝐶
9	8	nfcri 2888	. . . . . . . . 9 ⊢ Ⅎ𝑗 𝑥 ∈ 𝐶
10	7, 9	nfan 1900	. . . . . . . 8 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑥 ∈ 𝐶)
11		nfcv 2896	. . . . . . . . 9 ⊢ Ⅎ𝑗𝑦
12		nfmpt1 5195	. . . . . . . . . 10 ⊢ Ⅎ𝑗(𝑗 ∈ 𝐴 ↦ 𝐵)
13	12	nfrn 5899	. . . . . . . . 9 ⊢ Ⅎ𝑗ran (𝑗 ∈ 𝐴 ↦ 𝐵)
14	11, 13	nfel 2911	. . . . . . . 8 ⊢ Ⅎ𝑗 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)
15	10, 14	nfan 1900	. . . . . . 7 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵))
16		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑗 𝑥 ∈ 𝑦
17	15, 16	nfan 1900	. . . . . 6 ⊢ Ⅎ𝑗(((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑥 ∈ 𝑦)
18		simpllr 775	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑥 ∈ 𝑦) ∧ 𝑗 ∈ 𝐴) ∧ 𝑦 = 𝐵) → 𝑥 ∈ 𝑦)
19		simpr 484	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑥 ∈ 𝑦) ∧ 𝑗 ∈ 𝐴) ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
20	18, 19	eleqtrd 2836	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑥 ∈ 𝑦) ∧ 𝑗 ∈ 𝐴) ∧ 𝑦 = 𝐵) → 𝑥 ∈ 𝐵)
21	20	ex 412	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑥 ∈ 𝑦) ∧ 𝑗 ∈ 𝐴) → (𝑦 = 𝐵 → 𝑥 ∈ 𝐵))
22	21	ex 412	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑥 ∈ 𝑦) → (𝑗 ∈ 𝐴 → (𝑦 = 𝐵 → 𝑥 ∈ 𝐵)))
23	17, 22	reximdai 3236	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑥 ∈ 𝑦) → (∃𝑗 ∈ 𝐴 𝑦 = 𝐵 → ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵))
24	6, 23	mpd 15	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑥 ∈ 𝑦) → ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵)
25		acunirnmpt2f.2	. . . . . . . 8 ⊢ 𝐶 = ∪ 𝑗 ∈ 𝐴 𝐵
26		acunirnmpt2f.4	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ 𝑊)
27	26	ralrimiva 3126	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 𝐵 ∈ 𝑊)
28		dfiun3g 5915	. . . . . . . . 9 ⊢ (∀𝑗 ∈ 𝐴 𝐵 ∈ 𝑊 → ∪ 𝑗 ∈ 𝐴 𝐵 = ∪ ran (𝑗 ∈ 𝐴 ↦ 𝐵))
29	27, 28	syl 17	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑗 ∈ 𝐴 𝐵 = ∪ ran (𝑗 ∈ 𝐴 ↦ 𝐵))
30	25, 29	eqtrid 2781	. . . . . . 7 ⊢ (𝜑 → 𝐶 = ∪ ran (𝑗 ∈ 𝐴 ↦ 𝐵))
31	30	eleq2d 2820	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↔ 𝑥 ∈ ∪ ran (𝑗 ∈ 𝐴 ↦ 𝐵)))
32	31	biimpa 476	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ ∪ ran (𝑗 ∈ 𝐴 ↦ 𝐵))
33		eluni2 4865	. . . . 5 ⊢ (𝑥 ∈ ∪ ran (𝑗 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)𝑥 ∈ 𝑦)
34	32, 33	sylib 218	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∃𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)𝑥 ∈ 𝑦)
35	24, 34	r19.29a 3142	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵)
36	35	ralrimiva 3126	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵)
37		acunirnmpt.0	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
38		aciunf1lem.a	. . . . . . 7 ⊢ Ⅎ𝑗𝐴
39		nfcv 2896	. . . . . . 7 ⊢ Ⅎ𝑘𝐴
40		nfcv 2896	. . . . . . 7 ⊢ Ⅎ𝑘𝐵
41		nfcsb1v 3871	. . . . . . 7 ⊢ Ⅎ𝑗⦋𝑘 / 𝑗⦌𝐵
42		csbeq1a 3861	. . . . . . 7 ⊢ (𝑗 = 𝑘 → 𝐵 = ⦋𝑘 / 𝑗⦌𝐵)
43	38, 39, 40, 41, 42	cbvmptf 5196	. . . . . 6 ⊢ (𝑗 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ ⦋𝑘 / 𝑗⦌𝐵)
44		mptexg 7165	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑘 ∈ 𝐴 ↦ ⦋𝑘 / 𝑗⦌𝐵) ∈ V)
45	43, 44	eqeltrid 2838	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑗 ∈ 𝐴 ↦ 𝐵) ∈ V)
46		rnexg 7842	. . . . 5 ⊢ ((𝑗 ∈ 𝐴 ↦ 𝐵) ∈ V → ran (𝑗 ∈ 𝐴 ↦ 𝐵) ∈ V)
47		uniexg 7683	. . . . 5 ⊢ (ran (𝑗 ∈ 𝐴 ↦ 𝐵) ∈ V → ∪ ran (𝑗 ∈ 𝐴 ↦ 𝐵) ∈ V)
48	37, 45, 46, 47	4syl 19	. . . 4 ⊢ (𝜑 → ∪ ran (𝑗 ∈ 𝐴 ↦ 𝐵) ∈ V)
49	30, 48	eqeltrd 2834	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
50		id 22	. . . . . 6 ⊢ (𝑐 = 𝐶 → 𝑐 = 𝐶)
51	50	raleqdv 3294	. . . . 5 ⊢ (𝑐 = 𝐶 → (∀𝑥 ∈ 𝑐 ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐶 ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵))
52	50	feq2d 6644	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (𝑓:𝑐⟶𝐴 ↔ 𝑓:𝐶⟶𝐴))
53	50	raleqdv 3294	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (∀𝑥 ∈ 𝑐 𝑥 ∈ 𝐷 ↔ ∀𝑥 ∈ 𝐶 𝑥 ∈ 𝐷))
54	52, 53	anbi12d 632	. . . . . 6 ⊢ (𝑐 = 𝐶 → ((𝑓:𝑐⟶𝐴 ∧ ∀𝑥 ∈ 𝑐 𝑥 ∈ 𝐷) ↔ (𝑓:𝐶⟶𝐴 ∧ ∀𝑥 ∈ 𝐶 𝑥 ∈ 𝐷)))
55	54	exbidv 1922	. . . . 5 ⊢ (𝑐 = 𝐶 → (∃𝑓(𝑓:𝑐⟶𝐴 ∧ ∀𝑥 ∈ 𝑐 𝑥 ∈ 𝐷) ↔ ∃𝑓(𝑓:𝐶⟶𝐴 ∧ ∀𝑥 ∈ 𝐶 𝑥 ∈ 𝐷)))
56	51, 55	imbi12d 344	. . . 4 ⊢ (𝑐 = 𝐶 → ((∀𝑥 ∈ 𝑐 ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵 → ∃𝑓(𝑓:𝑐⟶𝐴 ∧ ∀𝑥 ∈ 𝑐 𝑥 ∈ 𝐷)) ↔ (∀𝑥 ∈ 𝐶 ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵 → ∃𝑓(𝑓:𝐶⟶𝐴 ∧ ∀𝑥 ∈ 𝐶 𝑥 ∈ 𝐷))))
57		acunirnmpt2f.d	. . . . . 6 ⊢ Ⅎ𝑗𝐷
58	57	nfcri 2888	. . . . 5 ⊢ Ⅎ𝑗 𝑥 ∈ 𝐷
59		vex 3442	. . . . 5 ⊢ 𝑐 ∈ V
60		acunirnmpt2f.3	. . . . . 6 ⊢ (𝑗 = (𝑓‘𝑥) → 𝐵 = 𝐷)
61	60	eleq2d 2820	. . . . 5 ⊢ (𝑗 = (𝑓‘𝑥) → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝐷))
62	38, 58, 59, 61	ac6sf2 32649	. . . 4 ⊢ (∀𝑥 ∈ 𝑐 ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵 → ∃𝑓(𝑓:𝑐⟶𝐴 ∧ ∀𝑥 ∈ 𝑐 𝑥 ∈ 𝐷))
63	56, 62	vtoclg 3509	. . 3 ⊢ (𝐶 ∈ V → (∀𝑥 ∈ 𝐶 ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵 → ∃𝑓(𝑓:𝐶⟶𝐴 ∧ ∀𝑥 ∈ 𝐶 𝑥 ∈ 𝐷)))
64	49, 63	syl 17	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐶 ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵 → ∃𝑓(𝑓:𝐶⟶𝐴 ∧ ∀𝑥 ∈ 𝐶 𝑥 ∈ 𝐷)))
65	36, 64	mpd 15	1 ⊢ (𝜑 → ∃𝑓(𝑓:𝐶⟶𝐴 ∧ ∀𝑥 ∈ 𝐶 𝑥 ∈ 𝐷))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  Ⅎwnfc 2881   ≠ wne 2930  ∀wral 3049  ∃wrex 3058  Vcvv 3438  ⦋csb 3847  ∅c0 4283  ∪ cuni 4861  ∪ ciun 4944   ↦ cmpt 5177  ran crn 5623  ⟶wf 6486  ‘cfv 6490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-reg 9495  ax-inf2 9548  ax-ac2 10371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-iin 4947  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-isom 6499  df-riota 7313  df-ov 7359  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-en 8882  df-r1 9674  df-rank 9675  df-card 9849  df-ac 10024

This theorem is referenced by:  aciunf1lem  32689