Theorem aciunf1 32637

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 4039	. . . 4 ⊢ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅} ⊆ 𝐴
2		aciunf1.0	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		ssexg 5273	. . . 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 3424	. . . . . 6 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅} ↔ (𝑗 ∈ 𝐴 ∧ 𝐵 ≠ ∅))
6	5	biimpi 216	. . . . 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 3423	. . 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 591	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}) → 𝐵 ∈ 𝑊)
13	4, 8, 9, 12	aciunf1lem 32636	. 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 2730	. . . . 5 ⊢ (𝜑 → 𝑓 = 𝑓)
15		nfv 1914	. . . . . . 7 ⊢ Ⅎ𝑗𝜑
16		nfcv 2891	. . . . . . . 8 ⊢ Ⅎ𝑗𝐴
17		nfrab1 3423	. . . . . . . 8 ⊢ Ⅎ𝑗{𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}
18	16, 17	nfdif 4088	. . . . . . 7 ⊢ Ⅎ𝑗(𝐴 ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅})
19		difrab 4277	. . . . . . . . 9 ⊢ ({𝑗 ∈ 𝐴 ∣ ⊤} ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}) = {𝑗 ∈ 𝐴 ∣ (⊤ ∧ ¬ 𝐵 = ∅)}
20	16	rabtru 3653	. . . . . . . . . 10 ⊢ {𝑗 ∈ 𝐴 ∣ ⊤} = 𝐴
21	20	difeq1i 4081	. . . . . . . . 9 ⊢ ({𝑗 ∈ 𝐴 ∣ ⊤} ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}) = (𝐴 ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅})
22		truan 1551	. . . . . . . . . . 11 ⊢ ((⊤ ∧ ¬ 𝐵 = ∅) ↔ ¬ 𝐵 = ∅)
23		df-ne 2926	. . . . . . . . . . 11 ⊢ (𝐵 ≠ ∅ ↔ ¬ 𝐵 = ∅)
24	22, 23	bitr4i 278	. . . . . . . . . 10 ⊢ ((⊤ ∧ ¬ 𝐵 = ∅) ↔ 𝐵 ≠ ∅)
25	24	rabbii 3408	. . . . . . . . 9 ⊢ {𝑗 ∈ 𝐴 ∣ (⊤ ∧ ¬ 𝐵 = ∅)} = {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}
26	19, 21, 25	3eqtr3i 2760	. . . . . . . 8 ⊢ (𝐴 ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}) = {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}
27	26	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}) = {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅})
28		eqidd 2730	. . . . . . 7 ⊢ (𝜑 → 𝐵 = 𝐵)
29	15, 18, 9, 27, 28	iuneq12df 4978	. . . . . 6 ⊢ (𝜑 → ∪ 𝑗 ∈ (𝐴 ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅})𝐵 = ∪ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}𝐵)
30		rabid 3424	. . . . . . . . . . 11 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅} ↔ (𝑗 ∈ 𝐴 ∧ 𝐵 = ∅))
31	30	biimpi 216	. . . . . . . . . 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 3125	. . . . . . 7 ⊢ (𝜑 → ∀𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}𝐵 = ∅)
35	17	iunxdif3 5054	. . . . . . 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 2766	. . . . 5 ⊢ (𝜑 → ∪ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}𝐵 = ∪ 𝑗 ∈ 𝐴 𝐵)
38		eqidd 2730	. . . . . . 7 ⊢ (𝜑 → ({𝑗} × 𝐵) = ({𝑗} × 𝐵))
39	15, 18, 9, 27, 38	iuneq12df 4978	. . . . . 6 ⊢ (𝜑 → ∪ 𝑗 ∈ (𝐴 ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅})({𝑗} × 𝐵) = ∪ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅} ({𝑗} × 𝐵))
40	33	xpeq2d 5661	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}) → ({𝑗} × 𝐵) = ({𝑗} × ∅))
41		xp0 6119	. . . . . . . . 9 ⊢ ({𝑗} × ∅) = ∅
42	40, 41	eqtrdi 2780	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}) → ({𝑗} × 𝐵) = ∅)
43	42	ralrimiva 3125	. . . . . . 7 ⊢ (𝜑 → ∀𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅} ({𝑗} × 𝐵) = ∅)
44	17	iunxdif3 5054	. . . . . . 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 2766	. . . . 5 ⊢ (𝜑 → ∪ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅} ({𝑗} × 𝐵) = ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
47	14, 37, 46	f1eq123d 6774	. . . 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 3296	. . . 4 ⊢ (𝜑 → (∀𝑘 ∈ ∪ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}𝐵(2nd ‘(𝑓‘𝑘)) = 𝑘 ↔ ∀𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑘)) = 𝑘))
49	47, 48	anbi12d 632	. . 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 1921	. 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 232	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 1540  ⊤wtru 1541  ∃wex 1779   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  {crab 3402  Vcvv 3444   ∖ cdif 3908   ⊆ wss 3911  ∅c0 4292  {csn 4585  ∪ ciun 4951   × cxp 5629  –1-1→wf1 6496  ‘cfv 6499  2^nd c2nd 7946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-reg 9521  ax-inf2 9570  ax-ac2 10392

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  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 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7326  df-ov 7372  df-om 7823  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-en 8896  df-r1 9693  df-rank 9694  df-card 9868  df-ac 10045

This theorem is referenced by:  fsumiunle  32804  esumiun  34077