Theorem acunirnmpt2f 32677

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 769	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑥 ∈ 𝑦) → 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵))
2		vex 3481	. . . . . . 7 ⊢ 𝑦 ∈ V
3		eqid 2734	. . . . . . . 8 ⊢ (𝑗 ∈ 𝐴 ↦ 𝐵) = (𝑗 ∈ 𝐴 ↦ 𝐵)
4	3	elrnmpt 5971	. . . . . . 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 1911	. . . . . . . . 9 ⊢ Ⅎ𝑗𝜑
8		acunirnmpt2f.c	. . . . . . . . . 10 ⊢ Ⅎ𝑗𝐶
9	8	nfcri 2894	. . . . . . . . 9 ⊢ Ⅎ𝑗 𝑥 ∈ 𝐶
10	7, 9	nfan 1896	. . . . . . . 8 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑥 ∈ 𝐶)
11		nfcv 2902	. . . . . . . . 9 ⊢ Ⅎ𝑗𝑦
12		nfmpt1 5255	. . . . . . . . . 10 ⊢ Ⅎ𝑗(𝑗 ∈ 𝐴 ↦ 𝐵)
13	12	nfrn 5965	. . . . . . . . 9 ⊢ Ⅎ𝑗ran (𝑗 ∈ 𝐴 ↦ 𝐵)
14	11, 13	nfel 2917	. . . . . . . 8 ⊢ Ⅎ𝑗 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)
15	10, 14	nfan 1896	. . . . . . 7 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵))
16		nfv 1911	. . . . . . 7 ⊢ Ⅎ𝑗 𝑥 ∈ 𝑦
17	15, 16	nfan 1896	. . . . . 6 ⊢ Ⅎ𝑗(((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑥 ∈ 𝑦)
18		simpllr 776	. . . . . . . . 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 2840	. . . . . . . 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 3258	. . . . 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 3143	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 𝐵 ∈ 𝑊)
28		dfiun3g 5980	. . . . . . . . 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 2786	. . . . . . 7 ⊢ (𝜑 → 𝐶 = ∪ ran (𝑗 ∈ 𝐴 ↦ 𝐵))
31	30	eleq2d 2824	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↔ 𝑥 ∈ ∪ ran (𝑗 ∈ 𝐴 ↦ 𝐵)))
32	31	biimpa 476	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ ∪ ran (𝑗 ∈ 𝐴 ↦ 𝐵))
33		eluni2 4915	. . . . 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 3159	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵)
36	35	ralrimiva 3143	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵)
37		acunirnmpt.0	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
38		aciunf1lem.a	. . . . . . 7 ⊢ Ⅎ𝑗𝐴
39		nfcv 2902	. . . . . . 7 ⊢ Ⅎ𝑘𝐴
40		nfcv 2902	. . . . . . 7 ⊢ Ⅎ𝑘𝐵
41		nfcsb1v 3932	. . . . . . 7 ⊢ Ⅎ𝑗⦋𝑘 / 𝑗⦌𝐵
42		csbeq1a 3921	. . . . . . 7 ⊢ (𝑗 = 𝑘 → 𝐵 = ⦋𝑘 / 𝑗⦌𝐵)
43	38, 39, 40, 41, 42	cbvmptf 5256	. . . . . 6 ⊢ (𝑗 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ ⦋𝑘 / 𝑗⦌𝐵)
44		mptexg 7240	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑘 ∈ 𝐴 ↦ ⦋𝑘 / 𝑗⦌𝐵) ∈ V)
45	43, 44	eqeltrid 2842	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑗 ∈ 𝐴 ↦ 𝐵) ∈ V)
46		rnexg 7924	. . . . 5 ⊢ ((𝑗 ∈ 𝐴 ↦ 𝐵) ∈ V → ran (𝑗 ∈ 𝐴 ↦ 𝐵) ∈ V)
47		uniexg 7758	. . . . 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 2838	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
50		id 22	. . . . . 6 ⊢ (𝑐 = 𝐶 → 𝑐 = 𝐶)
51	50	raleqdv 3323	. . . . 5 ⊢ (𝑐 = 𝐶 → (∀𝑥 ∈ 𝑐 ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐶 ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵))
52	50	feq2d 6722	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (𝑓:𝑐⟶𝐴 ↔ 𝑓:𝐶⟶𝐴))
53	50	raleqdv 3323	. . . . . . 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 1918	. . . . 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 2894	. . . . 5 ⊢ Ⅎ𝑗 𝑥 ∈ 𝐷
59		vex 3481	. . . . 5 ⊢ 𝑐 ∈ V
60		acunirnmpt2f.3	. . . . . 6 ⊢ (𝑗 = (𝑓‘𝑥) → 𝐵 = 𝐷)
61	60	eleq2d 2824	. . . . 5 ⊢ (𝑗 = (𝑓‘𝑥) → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝐷))
62	38, 58, 59, 61	ac6sf2 32641	. . . 4 ⊢ (∀𝑥 ∈ 𝑐 ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵 → ∃𝑓(𝑓:𝑐⟶𝐴 ∧ ∀𝑥 ∈ 𝑐 𝑥 ∈ 𝐷))
63	56, 62	vtoclg 3553	. . 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 1536  ∃wex 1775   ∈ wcel 2105  Ⅎwnfc 2887   ≠ wne 2937  ∀wral 3058  ∃wrex 3067  Vcvv 3477  ⦋csb 3907  ∅c0 4338  ∪ cuni 4911  ∪ ciun 4995   ↦ cmpt 5230  ran crn 5689  ⟶wf 6558  ‘cfv 6562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-reg 9629  ax-inf2 9678  ax-ac2 10500

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-en 8984  df-r1 9801  df-rank 9802  df-card 9976  df-ac 10153

This theorem is referenced by:  aciunf1lem  32678