Theorem acunirnmpt 30898

Description: Axiom of choice for the union of the range of a mapping to function. (Contributed by Thierry Arnoux, 6-Nov-2019.)

Hypotheses

Ref	Expression
acunirnmpt.0	⊢ (𝜑 → 𝐴 ∈ 𝑉)
acunirnmpt.1	⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ≠ ∅)
acunirnmpt.2	⊢ 𝐶 = ran (𝑗 ∈ 𝐴 ↦ 𝐵)

Assertion

Ref	Expression
acunirnmpt	⊢ (𝜑 → ∃𝑓(𝑓:𝐶⟶∪ 𝐶 ∧ ∀𝑦 ∈ 𝐶 ∃𝑗 ∈ 𝐴 (𝑓‘𝑦) ∈ 𝐵))

Distinct variable groups:   𝐴,𝑗   𝑓,𝑗,𝑦,𝐶   𝜑,𝑓,𝑗,𝑦

Allowed substitution hints:   𝐴(𝑦,𝑓)   𝐵(𝑦,𝑓,𝑗)   𝑉(𝑦,𝑓,𝑗)

Proof of Theorem acunirnmpt

Dummy variable 𝑐 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 484	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ 𝑗 ∈ 𝐴) ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
2		simplll 771	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ 𝑗 ∈ 𝐴) ∧ 𝑦 = 𝐵) → 𝜑)
3		simplr 765	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ 𝑗 ∈ 𝐴) ∧ 𝑦 = 𝐵) → 𝑗 ∈ 𝐴)
4		acunirnmpt.1	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ≠ ∅)
5	2, 3, 4	syl2anc 583	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ 𝑗 ∈ 𝐴) ∧ 𝑦 = 𝐵) → 𝐵 ≠ ∅)
6	1, 5	eqnetrd 3010	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ 𝑗 ∈ 𝐴) ∧ 𝑦 = 𝐵) → 𝑦 ≠ ∅)
7		acunirnmpt.2	. . . . . . . . 9 ⊢ 𝐶 = ran (𝑗 ∈ 𝐴 ↦ 𝐵)
8	7	eleq2i 2830	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵))
9		vex 3426	. . . . . . . . 9 ⊢ 𝑦 ∈ V
10		eqid 2738	. . . . . . . . . 10 ⊢ (𝑗 ∈ 𝐴 ↦ 𝐵) = (𝑗 ∈ 𝐴 ↦ 𝐵)
11	10	elrnmpt 5854	. . . . . . . . 9 ⊢ (𝑦 ∈ V → (𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑗 ∈ 𝐴 𝑦 = 𝐵))
12	9, 11	ax-mp 5	. . . . . . . 8 ⊢ (𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵) ↔ ∃𝑗 ∈ 𝐴 𝑦 = 𝐵)
13	8, 12	bitri 274	. . . . . . 7 ⊢ (𝑦 ∈ 𝐶 ↔ ∃𝑗 ∈ 𝐴 𝑦 = 𝐵)
14	13	biimpi 215	. . . . . 6 ⊢ (𝑦 ∈ 𝐶 → ∃𝑗 ∈ 𝐴 𝑦 = 𝐵)
15	14	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ∃𝑗 ∈ 𝐴 𝑦 = 𝐵)
16	6, 15	r19.29a 3217	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝑦 ≠ ∅)
17	16	ralrimiva 3107	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐶 𝑦 ≠ ∅)
18		acunirnmpt.0	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
19		mptexg 7079	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑗 ∈ 𝐴 ↦ 𝐵) ∈ V)
20		rnexg 7725	. . . . . 6 ⊢ ((𝑗 ∈ 𝐴 ↦ 𝐵) ∈ V → ran (𝑗 ∈ 𝐴 ↦ 𝐵) ∈ V)
21	18, 19, 20	3syl 18	. . . . 5 ⊢ (𝜑 → ran (𝑗 ∈ 𝐴 ↦ 𝐵) ∈ V)
22	7, 21	eqeltrid 2843	. . . 4 ⊢ (𝜑 → 𝐶 ∈ V)
23		raleq 3333	. . . . . 6 ⊢ (𝑐 = 𝐶 → (∀𝑦 ∈ 𝑐 𝑦 ≠ ∅ ↔ ∀𝑦 ∈ 𝐶 𝑦 ≠ ∅))
24		id 22	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → 𝑐 = 𝐶)
25		unieq 4847	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → ∪ 𝑐 = ∪ 𝐶)
26	24, 25	feq23d 6579	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (𝑓:𝑐⟶∪ 𝑐 ↔ 𝑓:𝐶⟶∪ 𝐶))
27		raleq 3333	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (∀𝑦 ∈ 𝑐 (𝑓‘𝑦) ∈ 𝑦 ↔ ∀𝑦 ∈ 𝐶 (𝑓‘𝑦) ∈ 𝑦))
28	26, 27	anbi12d 630	. . . . . . 7 ⊢ (𝑐 = 𝐶 → ((𝑓:𝑐⟶∪ 𝑐 ∧ ∀𝑦 ∈ 𝑐 (𝑓‘𝑦) ∈ 𝑦) ↔ (𝑓:𝐶⟶∪ 𝐶 ∧ ∀𝑦 ∈ 𝐶 (𝑓‘𝑦) ∈ 𝑦)))
29	28	exbidv 1925	. . . . . 6 ⊢ (𝑐 = 𝐶 → (∃𝑓(𝑓:𝑐⟶∪ 𝑐 ∧ ∀𝑦 ∈ 𝑐 (𝑓‘𝑦) ∈ 𝑦) ↔ ∃𝑓(𝑓:𝐶⟶∪ 𝐶 ∧ ∀𝑦 ∈ 𝐶 (𝑓‘𝑦) ∈ 𝑦)))
30	23, 29	imbi12d 344	. . . . 5 ⊢ (𝑐 = 𝐶 → ((∀𝑦 ∈ 𝑐 𝑦 ≠ ∅ → ∃𝑓(𝑓:𝑐⟶∪ 𝑐 ∧ ∀𝑦 ∈ 𝑐 (𝑓‘𝑦) ∈ 𝑦)) ↔ (∀𝑦 ∈ 𝐶 𝑦 ≠ ∅ → ∃𝑓(𝑓:𝐶⟶∪ 𝐶 ∧ ∀𝑦 ∈ 𝐶 (𝑓‘𝑦) ∈ 𝑦))))
31		vex 3426	. . . . . 6 ⊢ 𝑐 ∈ V
32	31	ac5b 10165	. . . . 5 ⊢ (∀𝑦 ∈ 𝑐 𝑦 ≠ ∅ → ∃𝑓(𝑓:𝑐⟶∪ 𝑐 ∧ ∀𝑦 ∈ 𝑐 (𝑓‘𝑦) ∈ 𝑦))
33	30, 32	vtoclg 3495	. . . 4 ⊢ (𝐶 ∈ V → (∀𝑦 ∈ 𝐶 𝑦 ≠ ∅ → ∃𝑓(𝑓:𝐶⟶∪ 𝐶 ∧ ∀𝑦 ∈ 𝐶 (𝑓‘𝑦) ∈ 𝑦)))
34	22, 33	syl 17	. . 3 ⊢ (𝜑 → (∀𝑦 ∈ 𝐶 𝑦 ≠ ∅ → ∃𝑓(𝑓:𝐶⟶∪ 𝐶 ∧ ∀𝑦 ∈ 𝐶 (𝑓‘𝑦) ∈ 𝑦)))
35	17, 34	mpd 15	. 2 ⊢ (𝜑 → ∃𝑓(𝑓:𝐶⟶∪ 𝐶 ∧ ∀𝑦 ∈ 𝐶 (𝑓‘𝑦) ∈ 𝑦))
36	15	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ (𝑓‘𝑦) ∈ 𝑦) → ∃𝑗 ∈ 𝐴 𝑦 = 𝐵)
37		simpllr 772	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ (𝑓‘𝑦) ∈ 𝑦) ∧ 𝑗 ∈ 𝐴) ∧ 𝑦 = 𝐵) → (𝑓‘𝑦) ∈ 𝑦)
38		simpr 484	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ (𝑓‘𝑦) ∈ 𝑦) ∧ 𝑗 ∈ 𝐴) ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
39	37, 38	eleqtrd 2841	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ (𝑓‘𝑦) ∈ 𝑦) ∧ 𝑗 ∈ 𝐴) ∧ 𝑦 = 𝐵) → (𝑓‘𝑦) ∈ 𝐵)
40	39	ex 412	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ (𝑓‘𝑦) ∈ 𝑦) ∧ 𝑗 ∈ 𝐴) → (𝑦 = 𝐵 → (𝑓‘𝑦) ∈ 𝐵))
41	40	reximdva 3202	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ (𝑓‘𝑦) ∈ 𝑦) → (∃𝑗 ∈ 𝐴 𝑦 = 𝐵 → ∃𝑗 ∈ 𝐴 (𝑓‘𝑦) ∈ 𝐵))
42	36, 41	mpd 15	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ (𝑓‘𝑦) ∈ 𝑦) → ∃𝑗 ∈ 𝐴 (𝑓‘𝑦) ∈ 𝐵)
43	42	ex 412	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ((𝑓‘𝑦) ∈ 𝑦 → ∃𝑗 ∈ 𝐴 (𝑓‘𝑦) ∈ 𝐵))
44	43	ralimdva 3102	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ 𝐶 (𝑓‘𝑦) ∈ 𝑦 → ∀𝑦 ∈ 𝐶 ∃𝑗 ∈ 𝐴 (𝑓‘𝑦) ∈ 𝐵))
45	44	anim2d 611	. . 3 ⊢ (𝜑 → ((𝑓:𝐶⟶∪ 𝐶 ∧ ∀𝑦 ∈ 𝐶 (𝑓‘𝑦) ∈ 𝑦) → (𝑓:𝐶⟶∪ 𝐶 ∧ ∀𝑦 ∈ 𝐶 ∃𝑗 ∈ 𝐴 (𝑓‘𝑦) ∈ 𝐵)))
46	45	eximdv 1921	. 2 ⊢ (𝜑 → (∃𝑓(𝑓:𝐶⟶∪ 𝐶 ∧ ∀𝑦 ∈ 𝐶 (𝑓‘𝑦) ∈ 𝑦) → ∃𝑓(𝑓:𝐶⟶∪ 𝐶 ∧ ∀𝑦 ∈ 𝐶 ∃𝑗 ∈ 𝐴 (𝑓‘𝑦) ∈ 𝐵)))
47	35, 46	mpd 15	1 ⊢ (𝜑 → ∃𝑓(𝑓:𝐶⟶∪ 𝐶 ∧ ∀𝑦 ∈ 𝐶 ∃𝑗 ∈ 𝐴 (𝑓‘𝑦) ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  Vcvv 3422  ∅c0 4253  ∪ cuni 4836   ↦ cmpt 5153  ran crn 5581  ⟶wf 6414  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-ac2 10150

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-en 8692  df-card 9628  df-ac 9803

This theorem is referenced by: (None)