Theorem acunirnmpt 32732

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 775	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ 𝑗 ∈ 𝐴) ∧ 𝑦 = 𝐵) → 𝜑)
3		simplr 769	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ 𝑗 ∈ 𝐴) ∧ 𝑦 = 𝐵) → 𝑗 ∈ 𝐴)
4		acunirnmpt.1	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ≠ ∅)
5	2, 3, 4	syl2anc 585	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ 𝑗 ∈ 𝐴) ∧ 𝑦 = 𝐵) → 𝐵 ≠ ∅)
6	1, 5	eqnetrd 2999	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ 𝐶) ∧ 𝑗 ∈ 𝐴) ∧ 𝑦 = 𝐵) → 𝑦 ≠ ∅)
7		acunirnmpt.2	. . . . . . . . 9 ⊢ 𝐶 = ran (𝑗 ∈ 𝐴 ↦ 𝐵)
8	7	eleq2i 2828	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ ran (𝑗 ∈ 𝐴 ↦ 𝐵))
9		vex 3433	. . . . . . . . 9 ⊢ 𝑦 ∈ V
10		eqid 2736	. . . . . . . . . 10 ⊢ (𝑗 ∈ 𝐴 ↦ 𝐵) = (𝑗 ∈ 𝐴 ↦ 𝐵)
11	10	elrnmpt 5913	. . . . . . . . 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 275	. . . . . . 7 ⊢ (𝑦 ∈ 𝐶 ↔ ∃𝑗 ∈ 𝐴 𝑦 = 𝐵)
14	13	biimpi 216	. . . . . 6 ⊢ (𝑦 ∈ 𝐶 → ∃𝑗 ∈ 𝐴 𝑦 = 𝐵)
15	14	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ∃𝑗 ∈ 𝐴 𝑦 = 𝐵)
16	6, 15	r19.29a 3145	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝑦 ≠ ∅)
17	16	ralrimiva 3129	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐶 𝑦 ≠ ∅)
18		acunirnmpt.0	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
19		mptexg 7176	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑗 ∈ 𝐴 ↦ 𝐵) ∈ V)
20		rnexg 7853	. . . . . 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 2840	. . . 4 ⊢ (𝜑 → 𝐶 ∈ V)
23		raleq 3292	. . . . . 6 ⊢ (𝑐 = 𝐶 → (∀𝑦 ∈ 𝑐 𝑦 ≠ ∅ ↔ ∀𝑦 ∈ 𝐶 𝑦 ≠ ∅))
24		id 22	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → 𝑐 = 𝐶)
25		unieq 4861	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → ∪ 𝑐 = ∪ 𝐶)
26	24, 25	feq23d 6663	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (𝑓:𝑐⟶∪ 𝑐 ↔ 𝑓:𝐶⟶∪ 𝐶))
27		raleq 3292	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (∀𝑦 ∈ 𝑐 (𝑓‘𝑦) ∈ 𝑦 ↔ ∀𝑦 ∈ 𝐶 (𝑓‘𝑦) ∈ 𝑦))
28	26, 27	anbi12d 633	. . . . . . 7 ⊢ (𝑐 = 𝐶 → ((𝑓:𝑐⟶∪ 𝑐 ∧ ∀𝑦 ∈ 𝑐 (𝑓‘𝑦) ∈ 𝑦) ↔ (𝑓:𝐶⟶∪ 𝐶 ∧ ∀𝑦 ∈ 𝐶 (𝑓‘𝑦) ∈ 𝑦)))
29	28	exbidv 1923	. . . . . 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 3433	. . . . . 6 ⊢ 𝑐 ∈ V
32	31	ac5b 10400	. . . . 5 ⊢ (∀𝑦 ∈ 𝑐 𝑦 ≠ ∅ → ∃𝑓(𝑓:𝑐⟶∪ 𝑐 ∧ ∀𝑦 ∈ 𝑐 (𝑓‘𝑦) ∈ 𝑦))
33	30, 32	vtoclg 3499	. . . 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 776	. . . . . . . . . 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 2838	. . . . . . . . 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 3150	. . . . . . 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 3149	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ 𝐶 (𝑓‘𝑦) ∈ 𝑦 → ∀𝑦 ∈ 𝐶 ∃𝑗 ∈ 𝐴 (𝑓‘𝑦) ∈ 𝐵))
45	44	anim2d 613	. . 3 ⊢ (𝜑 → ((𝑓:𝐶⟶∪ 𝐶 ∧ ∀𝑦 ∈ 𝐶 (𝑓‘𝑦) ∈ 𝑦) → (𝑓:𝐶⟶∪ 𝐶 ∧ ∀𝑦 ∈ 𝐶 ∃𝑗 ∈ 𝐴 (𝑓‘𝑦) ∈ 𝐵)))
46	45	eximdv 1919	. 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 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051  ∃wrex 3061  Vcvv 3429  ∅c0 4273  ∪ cuni 4850   ↦ cmpt 5166  ran crn 5632  ⟶wf 6494  ‘cfv 6498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-ac2 10385

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-en 8894  df-card 9863  df-ac 10038

This theorem is referenced by: (None)