Theorem aciunf1 30902

Description: Choice in an index union. (Contributed by Thierry Arnoux, 4-May-2020.)

Hypotheses

Ref	Expression
aciunf1.0	⊢ (𝜑 → 𝐴 ∈ 𝑉)
aciunf1.1	⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ 𝑊)

Assertion

Ref	Expression
aciunf1	⊢ (𝜑 → ∃𝑓(𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∧ ∀𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑘)) = 𝑘))

Distinct variable groups:   𝐴,𝑗,𝑘,𝑓   𝐵,𝑓,𝑘   𝑗,𝑊   𝜑,𝑓,𝑗,𝑘

Allowed substitution hints:   𝐵(𝑗)   𝑉(𝑓,𝑗,𝑘)   𝑊(𝑓,𝑘)

Proof of Theorem aciunf1

Step	Hyp	Ref	Expression
1		ssrab2 4009	. . . 4 ⊢ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅} ⊆ 𝐴
2		aciunf1.0	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		ssexg 5242	. . . 4 ⊢ (({𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅} ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅} ∈ V)
4	1, 2, 3	sylancr 586	. . 3 ⊢ (𝜑 → {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅} ∈ V)
5		rabid 3304	. . . . . 6 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅} ↔ (𝑗 ∈ 𝐴 ∧ 𝐵 ≠ ∅))
6	5	biimpi 215	. . . . 5 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅} → (𝑗 ∈ 𝐴 ∧ 𝐵 ≠ ∅))
7	6	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}) → (𝑗 ∈ 𝐴 ∧ 𝐵 ≠ ∅))
8	7	simprd 495	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}) → 𝐵 ≠ ∅)
9		nfrab1 3310	. . 3 ⊢ Ⅎ𝑗{𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}
10	7	simpld 494	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}) → 𝑗 ∈ 𝐴)
11		aciunf1.1	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ 𝑊)
12	10, 11	syldan 590	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}) → 𝐵 ∈ 𝑊)
13	4, 8, 9, 12	aciunf1lem 30901	. 2 ⊢ (𝜑 → ∃𝑓(𝑓:∪ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}𝐵–1-1→∪ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅} ({𝑗} × 𝐵) ∧ ∀𝑘 ∈ ∪ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}𝐵(2nd ‘(𝑓‘𝑘)) = 𝑘))
14		eqidd 2739	. . . . 5 ⊢ (𝜑 → 𝑓 = 𝑓)
15		nfv 1918	. . . . . . 7 ⊢ Ⅎ𝑗𝜑
16		nfcv 2906	. . . . . . . 8 ⊢ Ⅎ𝑗𝐴
17		nfrab1 3310	. . . . . . . 8 ⊢ Ⅎ𝑗{𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}
18	16, 17	nfdif 4056	. . . . . . 7 ⊢ Ⅎ𝑗(𝐴 ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅})
19		difrab 4239	. . . . . . . . 9 ⊢ ({𝑗 ∈ 𝐴 ∣ ⊤} ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}) = {𝑗 ∈ 𝐴 ∣ (⊤ ∧ ¬ 𝐵 = ∅)}
20	16	rabtru 3614	. . . . . . . . . 10 ⊢ {𝑗 ∈ 𝐴 ∣ ⊤} = 𝐴
21	20	difeq1i 4049	. . . . . . . . 9 ⊢ ({𝑗 ∈ 𝐴 ∣ ⊤} ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}) = (𝐴 ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅})
22		truan 1550	. . . . . . . . . . 11 ⊢ ((⊤ ∧ ¬ 𝐵 = ∅) ↔ ¬ 𝐵 = ∅)
23		df-ne 2943	. . . . . . . . . . 11 ⊢ (𝐵 ≠ ∅ ↔ ¬ 𝐵 = ∅)
24	22, 23	bitr4i 277	. . . . . . . . . 10 ⊢ ((⊤ ∧ ¬ 𝐵 = ∅) ↔ 𝐵 ≠ ∅)
25	24	rabbii 3397	. . . . . . . . 9 ⊢ {𝑗 ∈ 𝐴 ∣ (⊤ ∧ ¬ 𝐵 = ∅)} = {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}
26	19, 21, 25	3eqtr3i 2774	. . . . . . . 8 ⊢ (𝐴 ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}) = {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}
27	26	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}) = {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅})
28		eqidd 2739	. . . . . . 7 ⊢ (𝜑 → 𝐵 = 𝐵)
29	15, 18, 9, 27, 28	iuneq12df 4947	. . . . . 6 ⊢ (𝜑 → ∪ 𝑗 ∈ (𝐴 ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅})𝐵 = ∪ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}𝐵)
30		rabid 3304	. . . . . . . . . . 11 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅} ↔ (𝑗 ∈ 𝐴 ∧ 𝐵 = ∅))
31	30	biimpi 215	. . . . . . . . . 10 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅} → (𝑗 ∈ 𝐴 ∧ 𝐵 = ∅))
32	31	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}) → (𝑗 ∈ 𝐴 ∧ 𝐵 = ∅))
33	32	simprd 495	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}) → 𝐵 = ∅)
34	33	ralrimiva 3107	. . . . . . 7 ⊢ (𝜑 → ∀𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}𝐵 = ∅)
35	17	iunxdif3 5020	. . . . . . 7 ⊢ (∀𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}𝐵 = ∅ → ∪ 𝑗 ∈ (𝐴 ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅})𝐵 = ∪ 𝑗 ∈ 𝐴 𝐵)
36	34, 35	syl 17	. . . . . 6 ⊢ (𝜑 → ∪ 𝑗 ∈ (𝐴 ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅})𝐵 = ∪ 𝑗 ∈ 𝐴 𝐵)
37	29, 36	eqtr3d 2780	. . . . 5 ⊢ (𝜑 → ∪ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}𝐵 = ∪ 𝑗 ∈ 𝐴 𝐵)
38		eqidd 2739	. . . . . . 7 ⊢ (𝜑 → ({𝑗} × 𝐵) = ({𝑗} × 𝐵))
39	15, 18, 9, 27, 38	iuneq12df 4947	. . . . . 6 ⊢ (𝜑 → ∪ 𝑗 ∈ (𝐴 ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅})({𝑗} × 𝐵) = ∪ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅} ({𝑗} × 𝐵))
40	33	xpeq2d 5610	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}) → ({𝑗} × 𝐵) = ({𝑗} × ∅))
41		xp0 6050	. . . . . . . . 9 ⊢ ({𝑗} × ∅) = ∅
42	40, 41	eqtrdi 2795	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}) → ({𝑗} × 𝐵) = ∅)
43	42	ralrimiva 3107	. . . . . . 7 ⊢ (𝜑 → ∀𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅} ({𝑗} × 𝐵) = ∅)
44	17	iunxdif3 5020	. . . . . . 7 ⊢ (∀𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅} ({𝑗} × 𝐵) = ∅ → ∪ 𝑗 ∈ (𝐴 ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅})({𝑗} × 𝐵) = ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
45	43, 44	syl 17	. . . . . 6 ⊢ (𝜑 → ∪ 𝑗 ∈ (𝐴 ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅})({𝑗} × 𝐵) = ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
46	39, 45	eqtr3d 2780	. . . . 5 ⊢ (𝜑 → ∪ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅} ({𝑗} × 𝐵) = ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
47	14, 37, 46	f1eq123d 6692	. . . 4 ⊢ (𝜑 → (𝑓:∪ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}𝐵–1-1→∪ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅} ({𝑗} × 𝐵) ↔ 𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)))
48	37	raleqdv 3339	. . . 4 ⊢ (𝜑 → (∀𝑘 ∈ ∪ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}𝐵(2nd ‘(𝑓‘𝑘)) = 𝑘 ↔ ∀𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑘)) = 𝑘))
49	47, 48	anbi12d 630	. . 3 ⊢ (𝜑 → ((𝑓:∪ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}𝐵–1-1→∪ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅} ({𝑗} × 𝐵) ∧ ∀𝑘 ∈ ∪ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}𝐵(2nd ‘(𝑓‘𝑘)) = 𝑘) ↔ (𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∧ ∀𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑘)) = 𝑘)))
50	49	exbidv 1925	. 2 ⊢ (𝜑 → (∃𝑓(𝑓:∪ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}𝐵–1-1→∪ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅} ({𝑗} × 𝐵) ∧ ∀𝑘 ∈ ∪ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}𝐵(2nd ‘(𝑓‘𝑘)) = 𝑘) ↔ ∃𝑓(𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∧ ∀𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑘)) = 𝑘)))
51	13, 50	mpbid 231	1 ⊢ (𝜑 → ∃𝑓(𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∧ ∀𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑘)) = 𝑘))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1539  ⊤wtru 1540  ∃wex 1783   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  {crab 3067  Vcvv 3422   ∖ cdif 3880   ⊆ wss 3883  ∅c0 4253  {csn 4558  ∪ ciun 4921   × cxp 5578  –1-1→wf1 6415  ‘cfv 6418  2^nd c2nd 7803

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-reg 9281  ax-inf2 9329  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-iin 4924  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-lim 6256  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-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-en 8692  df-r1 9453  df-rank 9454  df-card 9628  df-ac 9803

This theorem is referenced by:  fsumiunle  31045  esumiun  31962