Theorem aciunf1 31000

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 4013	. . . 4 ⊢ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅} ⊆ 𝐴
2		aciunf1.0	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		ssexg 5247	. . . 4 ⊢ (({𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅} ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅} ∈ V)
4	1, 2, 3	sylancr 587	. . 3 ⊢ (𝜑 → {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅} ∈ V)
5		rabid 3310	. . . . . 6 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅} ↔ (𝑗 ∈ 𝐴 ∧ 𝐵 ≠ ∅))
6	5	biimpi 215	. . . . 5 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅} → (𝑗 ∈ 𝐴 ∧ 𝐵 ≠ ∅))
7	6	adantl 482	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}) → (𝑗 ∈ 𝐴 ∧ 𝐵 ≠ ∅))
8	7	simprd 496	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}) → 𝐵 ≠ ∅)
9		nfrab1 3317	. . 3 ⊢ Ⅎ𝑗{𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}
10	7	simpld 495	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}) → 𝑗 ∈ 𝐴)
11		aciunf1.1	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ 𝑊)
12	10, 11	syldan 591	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}) → 𝐵 ∈ 𝑊)
13	4, 8, 9, 12	aciunf1lem 30999	. 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 1917	. . . . . . 7 ⊢ Ⅎ𝑗𝜑
16		nfcv 2907	. . . . . . . 8 ⊢ Ⅎ𝑗𝐴
17		nfrab1 3317	. . . . . . . 8 ⊢ Ⅎ𝑗{𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}
18	16, 17	nfdif 4060	. . . . . . 7 ⊢ Ⅎ𝑗(𝐴 ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅})
19		difrab 4242	. . . . . . . . 9 ⊢ ({𝑗 ∈ 𝐴 ∣ ⊤} ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}) = {𝑗 ∈ 𝐴 ∣ (⊤ ∧ ¬ 𝐵 = ∅)}
20	16	rabtru 3621	. . . . . . . . . 10 ⊢ {𝑗 ∈ 𝐴 ∣ ⊤} = 𝐴
21	20	difeq1i 4053	. . . . . . . . 9 ⊢ ({𝑗 ∈ 𝐴 ∣ ⊤} ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}) = (𝐴 ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅})
22		truan 1550	. . . . . . . . . . 11 ⊢ ((⊤ ∧ ¬ 𝐵 = ∅) ↔ ¬ 𝐵 = ∅)
23		df-ne 2944	. . . . . . . . . . 11 ⊢ (𝐵 ≠ ∅ ↔ ¬ 𝐵 = ∅)
24	22, 23	bitr4i 277	. . . . . . . . . 10 ⊢ ((⊤ ∧ ¬ 𝐵 = ∅) ↔ 𝐵 ≠ ∅)
25	24	rabbii 3408	. . . . . . . . 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 4950	. . . . . 6 ⊢ (𝜑 → ∪ 𝑗 ∈ (𝐴 ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅})𝐵 = ∪ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}𝐵)
30		rabid 3310	. . . . . . . . . . 11 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅} ↔ (𝑗 ∈ 𝐴 ∧ 𝐵 = ∅))
31	30	biimpi 215	. . . . . . . . . 10 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅} → (𝑗 ∈ 𝐴 ∧ 𝐵 = ∅))
32	31	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}) → (𝑗 ∈ 𝐴 ∧ 𝐵 = ∅))
33	32	simprd 496	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}) → 𝐵 = ∅)
34	33	ralrimiva 3103	. . . . . . 7 ⊢ (𝜑 → ∀𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}𝐵 = ∅)
35	17	iunxdif3 5024	. . . . . . 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 4950	. . . . . 6 ⊢ (𝜑 → ∪ 𝑗 ∈ (𝐴 ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅})({𝑗} × 𝐵) = ∪ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅} ({𝑗} × 𝐵))
40	33	xpeq2d 5619	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}) → ({𝑗} × 𝐵) = ({𝑗} × ∅))
41		xp0 6061	. . . . . . . . 9 ⊢ ({𝑗} × ∅) = ∅
42	40, 41	eqtrdi 2794	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}) → ({𝑗} × 𝐵) = ∅)
43	42	ralrimiva 3103	. . . . . . 7 ⊢ (𝜑 → ∀𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅} ({𝑗} × 𝐵) = ∅)
44	17	iunxdif3 5024	. . . . . . 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 6708	. . . 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 3348	. . . 4 ⊢ (𝜑 → (∀𝑘 ∈ ∪ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}𝐵(2nd ‘(𝑓‘𝑘)) = 𝑘 ↔ ∀𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑘)) = 𝑘))
49	47, 48	anbi12d 631	. . 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 1924	. 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 396   = wceq 1539  ⊤wtru 1540  ∃wex 1782   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  {crab 3068  Vcvv 3432   ∖ cdif 3884   ⊆ wss 3887  ∅c0 4256  {csn 4561  ∪ ciun 4924   × cxp 5587  –1-1→wf1 6430  ‘cfv 6433  2^nd c2nd 7830

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-reg 9351  ax-inf2 9399  ax-ac2 10219

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-en 8734  df-r1 9522  df-rank 9523  df-card 9697  df-ac 9872

This theorem is referenced by:  fsumiunle  31143  esumiun  32062