Theorem aciunf1 32949

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 4042	. . . 4 ⊢ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅} ⊆ 𝐴
2		aciunf1.0	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		ssexg 5294	. . . 4 ⊢ (({𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅} ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅} ∈ V)
4	1, 2, 3	sylancr 598	. . 3 ⊢ (𝜑 → {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅} ∈ V)
5		rabid 3444	. . . . 5 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅} ↔ (𝑗 ∈ 𝐴 ∧ 𝐵 ≠ ∅))
6	5	bilani 509	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}) → (𝑗 ∈ 𝐴 ∧ 𝐵 ≠ ∅))
7	6	simprd 500	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}) → 𝐵 ≠ ∅)
8		nfrab1 3443	. . 3 ⊢ Ⅎ𝑗{𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}
9	6	simpld 499	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}) → 𝑗 ∈ 𝐴)
10		aciunf1.1	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ 𝑊)
11	9, 10	syldan 602	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}) → 𝐵 ∈ 𝑊)
12	4, 7, 8, 11	aciunf1lem 32948	. 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))
13		eqidd 2770	. . . . 5 ⊢ (𝜑 → 𝑓 = 𝑓)
14		nfv 1941	. . . . . . 7 ⊢ Ⅎ𝑗𝜑
15		nfcv 2931	. . . . . . . 8 ⊢ Ⅎ𝑗𝐴
16		nfrab1 3443	. . . . . . . 8 ⊢ Ⅎ𝑗{𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}
17	15, 16	nfdif 4092	. . . . . . 7 ⊢ Ⅎ𝑗(𝐴 ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅})
18		difrab 4279	. . . . . . . . 9 ⊢ ({𝑗 ∈ 𝐴 ∣ ⊤} ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}) = {𝑗 ∈ 𝐴 ∣ (⊤ ∧ ¬ 𝐵 = ∅)}
19	15	rabtru 3657	. . . . . . . . . 10 ⊢ {𝑗 ∈ 𝐴 ∣ ⊤} = 𝐴
20	19	difeq1i 4085	. . . . . . . . 9 ⊢ ({𝑗 ∈ 𝐴 ∣ ⊤} ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}) = (𝐴 ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅})
21		truan 1578	. . . . . . . . . . 11 ⊢ ((⊤ ∧ ¬ 𝐵 = ∅) ↔ ¬ 𝐵 = ∅)
22		df-ne 2965	. . . . . . . . . . 11 ⊢ (𝐵 ≠ ∅ ↔ ¬ 𝐵 = ∅)
23	21, 22	bitr4i 281	. . . . . . . . . 10 ⊢ ((⊤ ∧ ¬ 𝐵 = ∅) ↔ 𝐵 ≠ ∅)
24	23	rabbii 3428	. . . . . . . . 9 ⊢ {𝑗 ∈ 𝐴 ∣ (⊤ ∧ ¬ 𝐵 = ∅)} = {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}
25	18, 20, 24	3eqtr3i 2800	. . . . . . . 8 ⊢ (𝐴 ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}) = {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}
26	25	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}) = {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅})
27		eqidd 2770	. . . . . . 7 ⊢ (𝜑 → 𝐵 = 𝐵)
28	14, 17, 8, 26, 27	iuneq12df 4987	. . . . . 6 ⊢ (𝜑 → ∪ 𝑗 ∈ (𝐴 ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅})𝐵 = ∪ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}𝐵)
29		rabid 3444	. . . . . . . . . 10 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅} ↔ (𝑗 ∈ 𝐴 ∧ 𝐵 = ∅))
30	29	bilani 509	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}) → (𝑗 ∈ 𝐴 ∧ 𝐵 = ∅))
31	30	simprd 500	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}) → 𝐵 = ∅)
32	31	ralrimiva 3163	. . . . . . 7 ⊢ (𝜑 → ∀𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}𝐵 = ∅)
33	16	iunxdif3 5065	. . . . . . 7 ⊢ (∀𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}𝐵 = ∅ → ∪ 𝑗 ∈ (𝐴 ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅})𝐵 = ∪ 𝑗 ∈ 𝐴 𝐵)
34	32, 33	syl 18	. . . . . 6 ⊢ (𝜑 → ∪ 𝑗 ∈ (𝐴 ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅})𝐵 = ∪ 𝑗 ∈ 𝐴 𝐵)
35	28, 34	eqtr3d 2806	. . . . 5 ⊢ (𝜑 → ∪ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}𝐵 = ∪ 𝑗 ∈ 𝐴 𝐵)
36		eqidd 2770	. . . . . . 7 ⊢ (𝜑 → ({𝑗} × 𝐵) = ({𝑗} × 𝐵))
37	14, 17, 8, 26, 36	iuneq12df 4987	. . . . . 6 ⊢ (𝜑 → ∪ 𝑗 ∈ (𝐴 ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅})({𝑗} × 𝐵) = ∪ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅} ({𝑗} × 𝐵))
38	31	xpeq2d 5692	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}) → ({𝑗} × 𝐵) = ({𝑗} × ∅))
39		xp0 5762	. . . . . . . . 9 ⊢ ({𝑗} × ∅) = ∅
40	38, 39	eqtrdi 2820	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}) → ({𝑗} × 𝐵) = ∅)
41	40	ralrimiva 3163	. . . . . . 7 ⊢ (𝜑 → ∀𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅} ({𝑗} × 𝐵) = ∅)
42	16	iunxdif3 5065	. . . . . . 7 ⊢ (∀𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅} ({𝑗} × 𝐵) = ∅ → ∪ 𝑗 ∈ (𝐴 ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅})({𝑗} × 𝐵) = ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
43	41, 42	syl 18	. . . . . 6 ⊢ (𝜑 → ∪ 𝑗 ∈ (𝐴 ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅})({𝑗} × 𝐵) = ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
44	37, 43	eqtr3d 2806	. . . . 5 ⊢ (𝜑 → ∪ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅} ({𝑗} × 𝐵) = ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
45	13, 35, 44	f1eq123d 6813	. . . 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)))
46	35	raleqdv 3329	. . . 4 ⊢ (𝜑 → (∀𝑘 ∈ ∪ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}𝐵(2nd ‘(𝑓‘𝑘)) = 𝑘 ↔ ∀𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑘)) = 𝑘))
47	45, 46	anbi12d 643	. . 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)))
48	47	exbidv 1948	. 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)))
49	12, 48	mpbid 235	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 400   = wceq 1567  ⊤wtru 1568  ∃wex 1806   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  {crab 3423  Vcvv 3463   ∖ cdif 3910   ⊆ wss 3913  ∅c0 4294  {csn 4594  ∪ ciun 4960   × cxp 5660  –1-1→wf1 6534  ‘cfv 6537  2^nd c2nd 7985

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-reg 9554  ax-inf2 9610  ax-ac2 10447

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-iin 4963  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7368  df-ov 7414  df-om 7863  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-en 8944  df-r1 9736  df-rank 9737  df-card 9925  df-ac 10100

This theorem is referenced by:  fsumiunle  33114  esumiun  34429