Theorem aciunf1 32866

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 4034	. . . 4 ⊢ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅} ⊆ 𝐴
2		aciunf1.0	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		ssexg 5280	. . . 4 ⊢ (({𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅} ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅} ∈ V)
4	1, 2, 3	sylancr 596	. . 3 ⊢ (𝜑 → {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅} ∈ V)
5		rabid 3436	. . . . 5 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅} ↔ (𝑗 ∈ 𝐴 ∧ 𝐵 ≠ ∅))
6	5	bilani 508	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}) → (𝑗 ∈ 𝐴 ∧ 𝐵 ≠ ∅))
7	6	simprd 499	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}) → 𝐵 ≠ ∅)
8		nfrab1 3435	. . 3 ⊢ Ⅎ𝑗{𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}
9	6	simpld 498	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}) → 𝑗 ∈ 𝐴)
10		aciunf1.1	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ 𝑊)
11	9, 10	syldan 600	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}) → 𝐵 ∈ 𝑊)
12	4, 7, 8, 11	aciunf1lem 32865	. 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 2764	. . . . 5 ⊢ (𝜑 → 𝑓 = 𝑓)
14		nfv 1935	. . . . . . 7 ⊢ Ⅎ𝑗𝜑
15		nfcv 2925	. . . . . . . 8 ⊢ Ⅎ𝑗𝐴
16		nfrab1 3435	. . . . . . . 8 ⊢ Ⅎ𝑗{𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}
17	15, 16	nfdif 4084	. . . . . . 7 ⊢ Ⅎ𝑗(𝐴 ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅})
18		difrab 4271	. . . . . . . . 9 ⊢ ({𝑗 ∈ 𝐴 ∣ ⊤} ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}) = {𝑗 ∈ 𝐴 ∣ (⊤ ∧ ¬ 𝐵 = ∅)}
19	15	rabtru 3649	. . . . . . . . . 10 ⊢ {𝑗 ∈ 𝐴 ∣ ⊤} = 𝐴
20	19	difeq1i 4077	. . . . . . . . 9 ⊢ ({𝑗 ∈ 𝐴 ∣ ⊤} ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}) = (𝐴 ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅})
21		truan 1572	. . . . . . . . . . 11 ⊢ ((⊤ ∧ ¬ 𝐵 = ∅) ↔ ¬ 𝐵 = ∅)
22		df-ne 2959	. . . . . . . . . . 11 ⊢ (𝐵 ≠ ∅ ↔ ¬ 𝐵 = ∅)
23	21, 22	bitr4i 280	. . . . . . . . . 10 ⊢ ((⊤ ∧ ¬ 𝐵 = ∅) ↔ 𝐵 ≠ ∅)
24	23	rabbii 3420	. . . . . . . . 9 ⊢ {𝑗 ∈ 𝐴 ∣ (⊤ ∧ ¬ 𝐵 = ∅)} = {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}
25	18, 20, 24	3eqtr3i 2794	. . . . . . . 8 ⊢ (𝐴 ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}) = {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}
26	25	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}) = {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅})
27		eqidd 2764	. . . . . . 7 ⊢ (𝜑 → 𝐵 = 𝐵)
28	14, 17, 8, 26, 27	iuneq12df 4977	. . . . . 6 ⊢ (𝜑 → ∪ 𝑗 ∈ (𝐴 ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅})𝐵 = ∪ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}𝐵)
29		rabid 3436	. . . . . . . . . 10 ⊢ (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅} ↔ (𝑗 ∈ 𝐴 ∧ 𝐵 = ∅))
30	29	bilani 508	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}) → (𝑗 ∈ 𝐴 ∧ 𝐵 = ∅))
31	30	simprd 499	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}) → 𝐵 = ∅)
32	31	ralrimiva 3155	. . . . . . 7 ⊢ (𝜑 → ∀𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}𝐵 = ∅)
33	16	iunxdif3 5053	. . . . . . 7 ⊢ (∀𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}𝐵 = ∅ → ∪ 𝑗 ∈ (𝐴 ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅})𝐵 = ∪ 𝑗 ∈ 𝐴 𝐵)
34	32, 33	syl 17	. . . . . 6 ⊢ (𝜑 → ∪ 𝑗 ∈ (𝐴 ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅})𝐵 = ∪ 𝑗 ∈ 𝐴 𝐵)
35	28, 34	eqtr3d 2800	. . . . 5 ⊢ (𝜑 → ∪ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}𝐵 = ∪ 𝑗 ∈ 𝐴 𝐵)
36		eqidd 2764	. . . . . . 7 ⊢ (𝜑 → ({𝑗} × 𝐵) = ({𝑗} × 𝐵))
37	14, 17, 8, 26, 36	iuneq12df 4977	. . . . . 6 ⊢ (𝜑 → ∪ 𝑗 ∈ (𝐴 ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅})({𝑗} × 𝐵) = ∪ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅} ({𝑗} × 𝐵))
38	31	xpeq2d 5678	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}) → ({𝑗} × 𝐵) = ({𝑗} × ∅))
39		xp0 5748	. . . . . . . . 9 ⊢ ({𝑗} × ∅) = ∅
40	38, 39	eqtrdi 2814	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅}) → ({𝑗} × 𝐵) = ∅)
41	40	ralrimiva 3155	. . . . . . 7 ⊢ (𝜑 → ∀𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅} ({𝑗} × 𝐵) = ∅)
42	16	iunxdif3 5053	. . . . . . 7 ⊢ (∀𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅} ({𝑗} × 𝐵) = ∅ → ∪ 𝑗 ∈ (𝐴 ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅})({𝑗} × 𝐵) = ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
43	41, 42	syl 17	. . . . . 6 ⊢ (𝜑 → ∪ 𝑗 ∈ (𝐴 ∖ {𝑗 ∈ 𝐴 ∣ 𝐵 = ∅})({𝑗} × 𝐵) = ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
44	37, 43	eqtr3d 2800	. . . . 5 ⊢ (𝜑 → ∪ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅} ({𝑗} × 𝐵) = ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
45	13, 35, 44	f1eq123d 6799	. . . 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 3321	. . . 4 ⊢ (𝜑 → (∀𝑘 ∈ ∪ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐵 ≠ ∅}𝐵(2nd ‘(𝑓‘𝑘)) = 𝑘 ↔ ∀𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑘)) = 𝑘))
47	45, 46	anbi12d 641	. . 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 1942	. 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 234	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 399   = wceq 1561  ⊤wtru 1562  ∃wex 1800   ∈ wcel 2143   ≠ wne 2958  ∀wral 3077  {crab 3415  Vcvv 3455   ∖ cdif 3902   ⊆ wss 3905  ∅c0 4286  {csn 4583  ∪ ciun 4950   × cxp 5646  –1-1→wf1 6519  ‘cfv 6522  2^nd c2nd 7970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719  ax-reg 9541  ax-inf2 9597  ax-ac2 10421

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-se 5602  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6289  df-ord 6350  df-on 6351  df-lim 6352  df-suc 6353  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-f1 6527  df-fo 6528  df-f1o 6529  df-fv 6530  df-isom 6531  df-riota 7354  df-ov 7400  df-om 7848  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8382  df-en 8929  df-r1 9723  df-rank 9724  df-card 9898  df-ac 10073

This theorem is referenced by:  fsumiunle  33032  esumiun  34392