Theorem acunirnmpt2f 33312

Description: Axiom of choice for the union of the range of a mapping to function. (Contributed by Thierry Arnoux, 7-Nov-2019.) Remove dependency on ax-reg 9599 and ax-inf2 9655. (Revised by Timothy J. O'Malley, 15-Oct-2026.)

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 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		acunirnmpt2f.2	. . . . 5 ⊢ 𝐶 = ∪ 𝑗 ∈ 𝐴 𝐵
2		acunirnmpt2f.4	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ 𝑊)
3	2	ralrimiva 3158	. . . . . 6 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 𝐵 ∈ 𝑊)
4		dfiun3g 5955	. . . . . 6 ⊢ (∀𝑗 ∈ 𝐴 𝐵 ∈ 𝑊 → ∪ 𝑗 ∈ 𝐴 𝐵 = ∪ ran (𝑗 ∈ 𝐴 ↦ 𝐵))
5	3, 4	syl 18	. . . . 5 ⊢ (𝜑 → ∪ 𝑗 ∈ 𝐴 𝐵 = ∪ ran (𝑗 ∈ 𝐴 ↦ 𝐵))
6	1, 5	eqtrid 2810	. . . 4 ⊢ (𝜑 → 𝐶 = ∪ ran (𝑗 ∈ 𝐴 ↦ 𝐵))
7		acunirnmpt.0	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
8		aciunf1lem.a	. . . . . 6 ⊢ Ⅎ𝑗𝐴
9	8	mptexgf 7238	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑗 ∈ 𝐴 ↦ 𝐵) ∈ V)
10		rnexg 7924	. . . . 5 ⊢ ((𝑗 ∈ 𝐴 ↦ 𝐵) ∈ V → ran (𝑗 ∈ 𝐴 ↦ 𝐵) ∈ V)
11		uniexg 7767	. . . . 5 ⊢ (ran (𝑗 ∈ 𝐴 ↦ 𝐵) ∈ V → ∪ ran (𝑗 ∈ 𝐴 ↦ 𝐵) ∈ V)
12	7, 9, 10, 11	4syl 20	. . . 4 ⊢ (𝜑 → ∪ ran (𝑗 ∈ 𝐴 ↦ 𝐵) ∈ V)
13	6, 12	eqeltrd 2864	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
14	13, 7	jca 521	. 2 ⊢ (𝜑 → (𝐶 ∈ V ∧ 𝐴 ∈ 𝑉))
15		eqid 2763	. . . . . 6 ⊢ (𝑗 ∈ 𝐴 ↦ 𝐵) = (𝑗 ∈ 𝐴 ↦ 𝐵)
16		simplr 781	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑥 ∈ 𝑦) → 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵))
17	15, 16	elrnmpt2d 5953	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑥 ∈ 𝑦) → ∃𝑗 ∈ 𝐴 𝑦 = 𝐵)
18		nfv 1948	. . . . . . . . 9 ⊢ Ⅎ𝑗𝜑
19		acunirnmpt2f.c	. . . . . . . . . 10 ⊢ Ⅎ𝑗𝐶
20	19	nfcri 2918	. . . . . . . . 9 ⊢ Ⅎ𝑗 𝑥 ∈ 𝐶
21	18, 20	nfan 1933	. . . . . . . 8 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑥 ∈ 𝐶)
22		nfmpt1 5207	. . . . . . . . . 10 ⊢ Ⅎ𝑗(𝑗 ∈ 𝐴 ↦ 𝐵)
23	22	nfrn 5939	. . . . . . . . 9 ⊢ Ⅎ𝑗ran (𝑗 ∈ 𝐴 ↦ 𝐵)
24	23	nfcri 2918	. . . . . . . 8 ⊢ Ⅎ𝑗 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)
25	21, 24	nfan 1933	. . . . . . 7 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵))
26		nfv 1948	. . . . . . 7 ⊢ Ⅎ𝑗 𝑥 ∈ 𝑦
27	25, 26	nfan 1933	. . . . . 6 ⊢ Ⅎ𝑗(((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑥 ∈ 𝑦)
28		simpllr 788	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑥 ∈ 𝑦) ∧ 𝑗 ∈ 𝐴) ∧ 𝑦 = 𝐵) → 𝑥 ∈ 𝑦)
29		simpr 490	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑥 ∈ 𝑦) ∧ 𝑗 ∈ 𝐴) ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
30	28, 29	eleqtrd 2866	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑥 ∈ 𝑦) ∧ 𝑗 ∈ 𝐴) ∧ 𝑦 = 𝐵) → 𝑥 ∈ 𝐵)
31	30	exp31 425	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑥 ∈ 𝑦) → (𝑗 ∈ 𝐴 → (𝑦 = 𝐵 → 𝑥 ∈ 𝐵)))
32	27, 31	reximdai 3268	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑥 ∈ 𝑦) → (∃𝑗 ∈ 𝐴 𝑦 = 𝐵 → ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵))
33	17, 32	mpd 16	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)) ∧ 𝑥 ∈ 𝑦) → ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵)
34	6	eleq2d 2850	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↔ 𝑥 ∈ ∪ ran (𝑗 ∈ 𝐴 ↦ 𝐵)))
35	34	biimpa 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ ∪ ran (𝑗 ∈ 𝐴 ↦ 𝐵))
36		eluni2 4874	. . . . 5 ⊢ (𝑥 ∈ ∪ ran (𝑗 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)𝑥 ∈ 𝑦)
37	35, 36	sylib 221	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∃𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵)𝑥 ∈ 𝑦)
38	33, 37	r19.29a 3174	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵)
39	38	ralrimiva 3158	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵)
40		nfcv 2926	. . . . 5 ⊢ Ⅎ𝑐𝐴
41		nfv 1948	. . . . 5 ⊢ Ⅎ𝑐 𝑥 ∈ 𝐵
42		nfcsb1v 3875	. . . . . 6 ⊢ Ⅎ𝑗⦋𝑐 / 𝑗⦌𝐵
43	42	nfcri 2918	. . . . 5 ⊢ Ⅎ𝑗 𝑥 ∈ ⦋𝑐 / 𝑗⦌𝐵
44		csbeq1a 3865	. . . . . 6 ⊢ (𝑗 = 𝑐 → 𝐵 = ⦋𝑐 / 𝑗⦌𝐵)
45	44	eleq2d 2850	. . . . 5 ⊢ (𝑗 = 𝑐 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ ⦋𝑐 / 𝑗⦌𝐵))
46	8, 40, 41, 43, 45	cbvrexfw 3307	. . . 4 ⊢ (∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∃𝑐 ∈ 𝐴 𝑥 ∈ ⦋𝑐 / 𝑗⦌𝐵)
47	46	ralbii 3112	. . 3 ⊢ (∀𝑥 ∈ 𝐶 ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐶 ∃𝑐 ∈ 𝐴 𝑥 ∈ ⦋𝑐 / 𝑗⦌𝐵)
48		csbeq1 3854	. . . . . . 7 ⊢ (𝑐 = (𝑓‘𝑥) → ⦋𝑐 / 𝑗⦌𝐵 = ⦋(𝑓‘𝑥) / 𝑗⦌𝐵)
49		fvex 6909	. . . . . . . 8 ⊢ (𝑓‘𝑥) ∈ V
50		acunirnmpt2f.d	. . . . . . . 8 ⊢ Ⅎ𝑗𝐷
51		acunirnmpt2f.3	. . . . . . . 8 ⊢ (𝑗 = (𝑓‘𝑥) → 𝐵 = 𝐷)
52	49, 50, 51	csbief 3885	. . . . . . 7 ⊢ ⦋(𝑓‘𝑥) / 𝑗⦌𝐵 = 𝐷
53	48, 52	eqtrdi 2814	. . . . . 6 ⊢ (𝑐 = (𝑓‘𝑥) → ⦋𝑐 / 𝑗⦌𝐵 = 𝐷)
54	53	eleq2d 2850	. . . . 5 ⊢ (𝑐 = (𝑓‘𝑥) → (𝑥 ∈ ⦋𝑐 / 𝑗⦌𝐵 ↔ 𝑥 ∈ 𝐷))
55	54	ac6 10571	. . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐶 ∃𝑐 ∈ 𝐴 𝑥 ∈ ⦋𝑐 / 𝑗⦌𝐵) → ∃𝑓(𝑓:𝐶⟶𝐴 ∧ ∀𝑥 ∈ 𝐶 𝑥 ∈ 𝐷))
56	55	3expia 1139	. . 3 ⊢ ((𝐶 ∈ V ∧ 𝐴 ∈ 𝑉) → (∀𝑥 ∈ 𝐶 ∃𝑐 ∈ 𝐴 𝑥 ∈ ⦋𝑐 / 𝑗⦌𝐵 → ∃𝑓(𝑓:𝐶⟶𝐴 ∧ ∀𝑥 ∈ 𝐶 𝑥 ∈ 𝐷)))
57	47, 56	biimtrid 245	. 2 ⊢ ((𝐶 ∈ V ∧ 𝐴 ∈ 𝑉) → (∀𝑥 ∈ 𝐶 ∃𝑗 ∈ 𝐴 𝑥 ∈ 𝐵 → ∃𝑓(𝑓:𝐶⟶𝐴 ∧ ∀𝑥 ∈ 𝐶 𝑥 ∈ 𝐷)))
58	14, 39, 57	sylc 66	1 ⊢ (𝜑 → ∃𝑓(𝑓:𝐶⟶𝐴 ∧ ∀𝑥 ∈ 𝐶 𝑥 ∈ 𝐷))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2146  Ⅎwnfc 2911   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  Vcvv 3454  ⦋csb 3851  ∅c0 4281  ∪ cuni 4870  ∪ ciun 4954   ↦ cmpt 5189  ran crn 5656  ⟶wf 6543  ‘cfv 6547

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761  ax-ac2 10554

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  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 6311  df-ord 6372  df-on 6373  df-suc 6375  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-isom 6556  df-riota 7387  df-ov 7433  df-2nd 8012  df-frecs 8313  df-wrecs 8344  df-recs 8393  df-en 8988  df-card 10033  df-ac 10208

This theorem is used by:  aciunf1lem  33313