Theorem acunirnmpt2f 29461

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 792	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑥 ∈ 𝑦) → 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵))
2		vex 3203	. . . . . . 7 ⊢ 𝑦 ∈ V
3		eqid 2622	. . . . . . . 8 ⊢ (𝑗 ∈ 𝐴 ↦ 𝐵) = (𝑗 ∈ 𝐴 ↦ 𝐵)
4	3	elrnmpt 5372	. . . . . . 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 208	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑥 ∈ 𝑦) → ∃𝑗 ∈ 𝐴 𝑦 = 𝐵)
7		nfv 1843	. . . . . . . . 9 ⊢ Ⅎ𝑗𝜑
8		acunirnmpt2f.c	. . . . . . . . . 10 ⊢ Ⅎ𝑗𝐶
9	8	nfcri 2758	. . . . . . . . 9 ⊢ Ⅎ𝑗 𝑥 ∈ 𝐶
10	7, 9	nfan 1828	. . . . . . . 8 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑥 ∈ 𝐶)
11		nfcv 2764	. . . . . . . . 9 ⊢ Ⅎ𝑗𝑦
12		nfmpt1 4747	. . . . . . . . . 10 ⊢ Ⅎ𝑗(𝑗 ∈ 𝐴 ↦ 𝐵)
13	12	nfrn 5368	. . . . . . . . 9 ⊢ Ⅎ𝑗ran (𝑗 ∈ 𝐴 ↦ 𝐵)
14	11, 13	nfel 2777	. . . . . . . 8 ⊢ Ⅎ𝑗 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)
15	10, 14	nfan 1828	. . . . . . 7 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵))
16		nfv 1843	. . . . . . 7 ⊢ Ⅎ𝑗 𝑥 ∈ 𝑦
17	15, 16	nfan 1828	. . . . . 6 ⊢ Ⅎ𝑗(((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑥 ∈ 𝑦)
18		simpllr 799	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑥 ∈ 𝑦) ∧ 𝑗 ∈ 𝐴) ∧ 𝑦 = 𝐵) → 𝑥 ∈ 𝑦)
19		simpr 477	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑥 ∈ 𝑦) ∧ 𝑗 ∈ 𝐴) ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
20	18, 19	eleqtrd 2703	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑥 ∈ 𝑦) ∧ 𝑗 ∈ 𝐴) ∧ 𝑦 = 𝐵) → 𝑥 ∈ 𝐵)
21	20	ex 450	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑥 ∈ 𝑦) ∧ 𝑗 ∈ 𝐴) → (𝑦 = 𝐵 → 𝑥 ∈ 𝐵))
22	21	ex 450	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑥 ∈ 𝑦) → (𝑗 ∈ 𝐴 → (𝑦 = 𝐵 → 𝑥 ∈ 𝐵)))
23	17, 22	reximdai 3012	. . . . 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 2966	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 𝐵 ∈ 𝑊)
28		dfiun3g 5378	. . . . . . . . 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	syl5eq 2668	. . . . . . 7 ⊢ (𝜑 → 𝐶 = ∪ ran (𝑗 ∈ 𝐴 ↦ 𝐵))
31	30	eleq2d 2687	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↔ 𝑥 ∈ ∪ ran (𝑗 ∈ 𝐴 ↦ 𝐵)))
32	31	biimpa 501	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ ∪ ran (𝑗 ∈ 𝐴 ↦ 𝐵))
33		eluni2 4440	. . . . 5 ⊢ (𝑥 ∈ ∪ ran (𝑗 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)𝑥 ∈ 𝑦)
34	32, 33	sylib 208	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∃𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)𝑥 ∈ 𝑦)
35	24, 34	r19.29a 3078	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵)
36	35	ralrimiva 2966	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵)
37		acunirnmpt.0	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
38		aciunf1lem.a	. . . . . . 7 ⊢ Ⅎ𝑗𝐴
39		nfcv 2764	. . . . . . 7 ⊢ Ⅎ𝑘𝐴
40		nfcv 2764	. . . . . . 7 ⊢ Ⅎ𝑘𝐵
41		nfcsb1v 3549	. . . . . . 7 ⊢ Ⅎ𝑗⦋𝑘 / 𝑗⦌𝐵
42		csbeq1a 3542	. . . . . . 7 ⊢ (𝑗 = 𝑘 → 𝐵 = ⦋𝑘 / 𝑗⦌𝐵)
43	38, 39, 40, 41, 42	cbvmptf 4748	. . . . . 6 ⊢ (𝑗 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ ⦋𝑘 / 𝑗⦌𝐵)
44		mptexg 6484	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑘 ∈ 𝐴 ↦ ⦋𝑘 / 𝑗⦌𝐵) ∈ V)
45	43, 44	syl5eqel 2705	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑗 ∈ 𝐴 ↦ 𝐵) ∈ V)
46		rnexg 7098	. . . . 5 ⊢ ((𝑗 ∈ 𝐴 ↦ 𝐵) ∈ V → ran (𝑗 ∈ 𝐴 ↦ 𝐵) ∈ V)
47		uniexg 6955	. . . . 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 2701	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
50		id 22	. . . . . 6 ⊢ (𝑐 = 𝐶 → 𝑐 = 𝐶)
51	50	raleqdv 3144	. . . . 5 ⊢ (𝑐 = 𝐶 → (∀𝑥 ∈ 𝑐 ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐶 ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵))
52	50	feq2d 6031	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (𝑓:𝑐⟶𝐴 ↔ 𝑓:𝐶⟶𝐴))
53	50	raleqdv 3144	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (∀𝑥 ∈ 𝑐 𝑥 ∈ 𝐷 ↔ ∀𝑥 ∈ 𝐶 𝑥 ∈ 𝐷))
54	52, 53	anbi12d 747	. . . . . 6 ⊢ (𝑐 = 𝐶 → ((𝑓:𝑐⟶𝐴 ∧ ∀𝑥 ∈ 𝑐 𝑥 ∈ 𝐷) ↔ (𝑓:𝐶⟶𝐴 ∧ ∀𝑥 ∈ 𝐶 𝑥 ∈ 𝐷)))
55	54	exbidv 1850	. . . . 5 ⊢ (𝑐 = 𝐶 → (∃𝑓(𝑓:𝑐⟶𝐴 ∧ ∀𝑥 ∈ 𝑐 𝑥 ∈ 𝐷) ↔ ∃𝑓(𝑓:𝐶⟶𝐴 ∧ ∀𝑥 ∈ 𝐶 𝑥 ∈ 𝐷)))
56	51, 55	imbi12d 334	. . . 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 2758	. . . . 5 ⊢ Ⅎ𝑗 𝑥 ∈ 𝐷
59		vex 3203	. . . . 5 ⊢ 𝑐 ∈ V
60		acunirnmpt2f.3	. . . . . 6 ⊢ (𝑗 = (𝑓‘𝑥) → 𝐵 = 𝐷)
61	60	eleq2d 2687	. . . . 5 ⊢ (𝑗 = (𝑓‘𝑥) → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝐷))
62	38, 58, 59, 61	ac6sf2 29429	. . . 4 ⊢ (∀𝑥 ∈ 𝑐 ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵 → ∃𝑓(𝑓:𝑐⟶𝐴 ∧ ∀𝑥 ∈ 𝑐 𝑥 ∈ 𝐷))
63	56, 62	vtoclg 3266	. . 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 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  Ⅎwnfc 2751   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  Vcvv 3200  ⦋csb 3533  ∅c0 3915  ∪ cuni 4436  ∪ ciun 4520   ↦ cmpt 4729  ran crn 5115  ⟶wf 5884  ‘cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497  ax-inf2 8538  ax-ac2 9285

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-en 7956  df-r1 8627  df-rank 8628  df-card 8765  df-ac 8939

This theorem is referenced by:  aciunf1lem  29462