Theorem acunirnmpt 30992

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 485	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ 𝑗 ∈ 𝐴) ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
2		simplll 772	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ 𝑗 ∈ 𝐴) ∧ 𝑦 = 𝐵) → 𝜑)
3		simplr 766	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ 𝑗 ∈ 𝐴) ∧ 𝑦 = 𝐵) → 𝑗 ∈ 𝐴)
4		acunirnmpt.1	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ≠ ∅)
5	2, 3, 4	syl2anc 584	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ 𝑗 ∈ 𝐴) ∧ 𝑦 = 𝐵) → 𝐵 ≠ ∅)
6	1, 5	eqnetrd 3013	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ 𝑗 ∈ 𝐴) ∧ 𝑦 = 𝐵) → 𝑦 ≠ ∅)
7		acunirnmpt.2	. . . . . . . . 9 ⊢ 𝐶 = ran (𝑗 ∈ 𝐴 ↦ 𝐵)
8	7	eleq2i 2832	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵))
9		vex 3435	. . . . . . . . 9 ⊢ 𝑦 ∈ V
10		eqid 2740	. . . . . . . . . 10 ⊢ (𝑗 ∈ 𝐴 ↦ 𝐵) = (𝑗 ∈ 𝐴 ↦ 𝐵)
11	10	elrnmpt 5864	. . . . . . . . 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 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ∃𝑗 ∈ 𝐴 𝑦 = 𝐵)
16	6, 15	r19.29a 3220	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝑦 ≠ ∅)
17	16	ralrimiva 3110	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐶 𝑦 ≠ ∅)
18		acunirnmpt.0	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
19		mptexg 7094	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑗 ∈ 𝐴 ↦ 𝐵) ∈ V)
20		rnexg 7745	. . . . . 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 2845	. . . 4 ⊢ (𝜑 → 𝐶 ∈ V)
23		raleq 3341	. . . . . 6 ⊢ (𝑐 = 𝐶 → (∀𝑦 ∈ 𝑐 𝑦 ≠ ∅ ↔ ∀𝑦 ∈ 𝐶 𝑦 ≠ ∅))
24		id 22	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → 𝑐 = 𝐶)
25		unieq 4856	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → ∪ 𝑐 = ∪ 𝐶)
26	24, 25	feq23d 6593	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (𝑓:𝑐⟶∪ 𝑐 ↔ 𝑓:𝐶⟶∪ 𝐶))
27		raleq 3341	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (∀𝑦 ∈ 𝑐 (𝑓‘𝑦) ∈ 𝑦 ↔ ∀𝑦 ∈ 𝐶 (𝑓‘𝑦) ∈ 𝑦))
28	26, 27	anbi12d 631	. . . . . . 7 ⊢ (𝑐 = 𝐶 → ((𝑓:𝑐⟶∪ 𝑐 ∧ ∀𝑦 ∈ 𝑐 (𝑓‘𝑦) ∈ 𝑦) ↔ (𝑓:𝐶⟶∪ 𝐶 ∧ ∀𝑦 ∈ 𝐶 (𝑓‘𝑦) ∈ 𝑦)))
29	28	exbidv 1928	. . . . . 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 345	. . . . 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 3435	. . . . . 6 ⊢ 𝑐 ∈ V
32	31	ac5b 10235	. . . . 5 ⊢ (∀𝑦 ∈ 𝑐 𝑦 ≠ ∅ → ∃𝑓(𝑓:𝑐⟶∪ 𝑐 ∧ ∀𝑦 ∈ 𝑐 (𝑓‘𝑦) ∈ 𝑦))
33	30, 32	vtoclg 3504	. . . 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 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ (𝑓‘𝑦) ∈ 𝑦) → ∃𝑗 ∈ 𝐴 𝑦 = 𝐵)
37		simpllr 773	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ (𝑓‘𝑦) ∈ 𝑦) ∧ 𝑗 ∈ 𝐴) ∧ 𝑦 = 𝐵) → (𝑓‘𝑦) ∈ 𝑦)
38		simpr 485	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ (𝑓‘𝑦) ∈ 𝑦) ∧ 𝑗 ∈ 𝐴) ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
39	37, 38	eleqtrd 2843	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ (𝑓‘𝑦) ∈ 𝑦) ∧ 𝑗 ∈ 𝐴) ∧ 𝑦 = 𝐵) → (𝑓‘𝑦) ∈ 𝐵)
40	39	ex 413	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ (𝑓‘𝑦) ∈ 𝑦) ∧ 𝑗 ∈ 𝐴) → (𝑦 = 𝐵 → (𝑓‘𝑦) ∈ 𝐵))
41	40	reximdva 3205	. . . . . . 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 413	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ((𝑓‘𝑦) ∈ 𝑦 → ∃𝑗 ∈ 𝐴 (𝑓‘𝑦) ∈ 𝐵))
44	43	ralimdva 3105	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ 𝐶 (𝑓‘𝑦) ∈ 𝑦 → ∀𝑦 ∈ 𝐶 ∃𝑗 ∈ 𝐴 (𝑓‘𝑦) ∈ 𝐵))
45	44	anim2d 612	. . 3 ⊢ (𝜑 → ((𝑓:𝐶⟶∪ 𝐶 ∧ ∀𝑦 ∈ 𝐶 (𝑓‘𝑦) ∈ 𝑦) → (𝑓:𝐶⟶∪ 𝐶 ∧ ∀𝑦 ∈ 𝐶 ∃𝑗 ∈ 𝐴 (𝑓‘𝑦) ∈ 𝐵)))
46	45	eximdv 1924	. 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 396   = wceq 1542  ∃wex 1786   ∈ wcel 2110   ≠ wne 2945  ∀wral 3066  ∃wrex 3067  Vcvv 3431  ∅c0 4262  ∪ cuni 4845   ↦ cmpt 5162  ran crn 5591  ⟶wf 6428  ‘cfv 6432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582  ax-ac2 10220

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-int 4886  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-se 5546  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201  df-ord 6268  df-on 6269  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-isom 6441  df-riota 7228  df-ov 7274  df-2nd 7825  df-frecs 8088  df-wrecs 8119  df-recs 8193  df-en 8717  df-card 9698  df-ac 9873

This theorem is referenced by: (None)