Theorem acunirnmpt2 33136

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 ≠ ∅)
acunirnmpt2.2	⊢ 𝐶 = ∪ ran (𝑗 ∈ 𝐴 ↦ 𝐵)
acunirnmpt2.3	⊢ (𝑗 = (𝑓‘𝑥) → 𝐵 = 𝐷)

Assertion

Ref	Expression
acunirnmpt2	⊢ (𝜑 → ∃𝑓(𝑓:𝐶⟶𝐴 ∧ ∀𝑥 ∈ 𝐶 𝑥 ∈ 𝐷))

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

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

Proof of Theorem acunirnmpt2

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

Step	Hyp	Ref	Expression
1		simplr 781	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑥 ∈ 𝑦) → 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵))
2		vex 3454	. . . . . . 7 ⊢ 𝑦 ∈ V
3		eqid 2760	. . . . . . . 8 ⊢ (𝑗 ∈ 𝐴 ↦ 𝐵) = (𝑗 ∈ 𝐴 ↦ 𝐵)
4	3	elrnmpt 5942	. . . . . . 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 221	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑥 ∈ 𝑦) → ∃𝑗 ∈ 𝐴 𝑦 = 𝐵)
7		nfv 1947	. . . . . . . 8 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑥 ∈ 𝐶)
8		nfcv 2922	. . . . . . . . 9 ⊢ Ⅎ𝑗𝑦
9		nfmpt1 5204	. . . . . . . . . 10 ⊢ Ⅎ𝑗(𝑗 ∈ 𝐴 ↦ 𝐵)
10	9	nfrn 5936	. . . . . . . . 9 ⊢ Ⅎ𝑗ran (𝑗 ∈ 𝐴 ↦ 𝐵)
11	8, 10	nfel 2936	. . . . . . . 8 ⊢ Ⅎ𝑗 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)
12	7, 11	nfan 1932	. . . . . . 7 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵))
13		nfv 1947	. . . . . . 7 ⊢ Ⅎ𝑗 𝑥 ∈ 𝑦
14	12, 13	nfan 1932	. . . . . 6 ⊢ Ⅎ𝑗(((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑥 ∈ 𝑦)
15		simpllr 788	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑥 ∈ 𝑦) ∧ 𝑗 ∈ 𝐴) ∧ 𝑦 = 𝐵) → 𝑥 ∈ 𝑦)
16		simpr 490	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑥 ∈ 𝑦) ∧ 𝑗 ∈ 𝐴) ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
17	15, 16	eleqtrd 2862	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑥 ∈ 𝑦) ∧ 𝑗 ∈ 𝐴) ∧ 𝑦 = 𝐵) → 𝑥 ∈ 𝐵)
18	17	ex 418	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑥 ∈ 𝑦) ∧ 𝑗 ∈ 𝐴) → (𝑦 = 𝐵 → 𝑥 ∈ 𝐵))
19	18	ex 418	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑥 ∈ 𝑦) → (𝑗 ∈ 𝐴 → (𝑦 = 𝐵 → 𝑥 ∈ 𝐵)))
20	14, 19	reximdai 3264	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑥 ∈ 𝑦) → (∃𝑗 ∈ 𝐴 𝑦 = 𝐵 → ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵))
21	6, 20	mpd 16	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑥 ∈ 𝑦) → ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵)
22		acunirnmpt2.2	. . . . . . . 8 ⊢ 𝐶 = ∪ ran (𝑗 ∈ 𝐴 ↦ 𝐵)
23	22	eleq2i 2852	. . . . . . 7 ⊢ (𝑥 ∈ 𝐶 ↔ 𝑥 ∈ ∪ ran (𝑗 ∈ 𝐴 ↦ 𝐵))
24	23	biimpi 219	. . . . . 6 ⊢ (𝑥 ∈ 𝐶 → 𝑥 ∈ ∪ ran (𝑗 ∈ 𝐴 ↦ 𝐵))
25		eluni2 4871	. . . . . 6 ⊢ (𝑥 ∈ ∪ ran (𝑗 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)𝑥 ∈ 𝑦)
26	24, 25	sylib 221	. . . . 5 ⊢ (𝑥 ∈ 𝐶 → ∃𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)𝑥 ∈ 𝑦)
27	26	adantl 487	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∃𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)𝑥 ∈ 𝑦)
28	21, 27	r19.29a 3170	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵)
29	28	ralrimiva 3154	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵)
30		acunirnmpt.0	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
31		mptexg 7221	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑗 ∈ 𝐴 ↦ 𝐵) ∈ V)
32		rnexg 7900	. . . . 5 ⊢ ((𝑗 ∈ 𝐴 ↦ 𝐵) ∈ V → ran (𝑗 ∈ 𝐴 ↦ 𝐵) ∈ V)
33		uniexg 7743	. . . . 5 ⊢ (ran (𝑗 ∈ 𝐴 ↦ 𝐵) ∈ V → ∪ ran (𝑗 ∈ 𝐴 ↦ 𝐵) ∈ V)
34	30, 31, 32, 33	4syl 20	. . . 4 ⊢ (𝜑 → ∪ ran (𝑗 ∈ 𝐴 ↦ 𝐵) ∈ V)
35	22, 34	eqeltrid 2864	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
36		id 23	. . . . . 6 ⊢ (𝑐 = 𝐶 → 𝑐 = 𝐶)
37	36	raleqdv 3319	. . . . 5 ⊢ (𝑐 = 𝐶 → (∀𝑥 ∈ 𝑐 ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐶 ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵))
38	36	feq2d 6687	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (𝑓:𝑐⟶𝐴 ↔ 𝑓:𝐶⟶𝐴))
39	36	raleqdv 3319	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (∀𝑥 ∈ 𝑐 𝑥 ∈ 𝐷 ↔ ∀𝑥 ∈ 𝐶 𝑥 ∈ 𝐷))
40	38, 39	anbi12d 644	. . . . . 6 ⊢ (𝑐 = 𝐶 → ((𝑓:𝑐⟶𝐴 ∧ ∀𝑥 ∈ 𝑐 𝑥 ∈ 𝐷) ↔ (𝑓:𝐶⟶𝐴 ∧ ∀𝑥 ∈ 𝐶 𝑥 ∈ 𝐷)))
41	40	exbidv 1954	. . . . 5 ⊢ (𝑐 = 𝐶 → (∃𝑓(𝑓:𝑐⟶𝐴 ∧ ∀𝑥 ∈ 𝑐 𝑥 ∈ 𝐷) ↔ ∃𝑓(𝑓:𝐶⟶𝐴 ∧ ∀𝑥 ∈ 𝐶 𝑥 ∈ 𝐷)))
42	37, 41	imbi12d 347	. . . 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))))
43		vex 3454	. . . . 5 ⊢ 𝑐 ∈ V
44		acunirnmpt2.3	. . . . . 6 ⊢ (𝑗 = (𝑓‘𝑥) → 𝐵 = 𝐷)
45	44	eleq2d 2846	. . . . 5 ⊢ (𝑗 = (𝑓‘𝑥) → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝐷))
46	43, 45	ac6s 10489	. . . 4 ⊢ (∀𝑥 ∈ 𝑐 ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵 → ∃𝑓(𝑓:𝑐⟶𝐴 ∧ ∀𝑥 ∈ 𝑐 𝑥 ∈ 𝐷))
47	42, 46	vtoclg 3517	. . 3 ⊢ (𝐶 ∈ V → (∀𝑥 ∈ 𝐶 ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵 → ∃𝑓(𝑓:𝐶⟶𝐴 ∧ ∀𝑥 ∈ 𝐶 𝑥 ∈ 𝐷)))
48	35, 47	syl 18	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐶 ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵 → ∃𝑓(𝑓:𝐶⟶𝐴 ∧ ∀𝑥 ∈ 𝐶 𝑥 ∈ 𝐷)))
49	29, 48	mpd 16	1 ⊢ (𝜑 → ∃𝑓(𝑓:𝐶⟶𝐴 ∧ ∀𝑥 ∈ 𝐶 𝑥 ∈ 𝐷))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  Vcvv 3450  ∅c0 4279  ∪ cuni 4867   ↦ cmpt 5186  ran crn 5656  ⟶wf 6529  ‘cfv 6533

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737  ax-reg 9567  ax-inf2 9623  ax-ac2 10468

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7371  df-ov 7417  df-om 7864  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-en 8956  df-r1 9749  df-rank 9750  df-scott 9871  df-card 9947  df-ac 10122

This theorem is used by: (None)