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Theorem aciunf1 32155
Description: Choice in an index union. (Contributed by Thierry Arnoux, 4-May-2020.)
Hypotheses
Ref Expression
aciunf1.0 (𝜑𝐴𝑉)
aciunf1.1 ((𝜑𝑗𝐴) → 𝐵𝑊)
Assertion
Ref Expression
aciunf1 (𝜑 → ∃𝑓(𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵) ∧ ∀𝑘 𝑗𝐴 𝐵(2nd ‘(𝑓𝑘)) = 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
StepHypRef Expression
1 ssrab2 4076 . . . 4 {𝑗𝐴𝐵 ≠ ∅} ⊆ 𝐴
2 aciunf1.0 . . . 4 (𝜑𝐴𝑉)
3 ssexg 5322 . . . 4 (({𝑗𝐴𝐵 ≠ ∅} ⊆ 𝐴𝐴𝑉) → {𝑗𝐴𝐵 ≠ ∅} ∈ V)
41, 2, 3sylancr 585 . . 3 (𝜑 → {𝑗𝐴𝐵 ≠ ∅} ∈ V)
5 rabid 3450 . . . . . 6 (𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅} ↔ (𝑗𝐴𝐵 ≠ ∅))
65biimpi 215 . . . . 5 (𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅} → (𝑗𝐴𝐵 ≠ ∅))
76adantl 480 . . . 4 ((𝜑𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}) → (𝑗𝐴𝐵 ≠ ∅))
87simprd 494 . . 3 ((𝜑𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}) → 𝐵 ≠ ∅)
9 nfrab1 3449 . . 3 𝑗{𝑗𝐴𝐵 ≠ ∅}
107simpld 493 . . . 4 ((𝜑𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}) → 𝑗𝐴)
11 aciunf1.1 . . . 4 ((𝜑𝑗𝐴) → 𝐵𝑊)
1210, 11syldan 589 . . 3 ((𝜑𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}) → 𝐵𝑊)
134, 8, 9, 12aciunf1lem 32154 . 2 (𝜑 → ∃𝑓(𝑓: 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}𝐵1-1 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅} ({𝑗} × 𝐵) ∧ ∀𝑘 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}𝐵(2nd ‘(𝑓𝑘)) = 𝑘))
14 eqidd 2731 . . . . 5 (𝜑𝑓 = 𝑓)
15 nfv 1915 . . . . . . 7 𝑗𝜑
16 nfcv 2901 . . . . . . . 8 𝑗𝐴
17 nfrab1 3449 . . . . . . . 8 𝑗{𝑗𝐴𝐵 = ∅}
1816, 17nfdif 4124 . . . . . . 7 𝑗(𝐴 ∖ {𝑗𝐴𝐵 = ∅})
19 difrab 4307 . . . . . . . . 9 ({𝑗𝐴 ∣ ⊤} ∖ {𝑗𝐴𝐵 = ∅}) = {𝑗𝐴 ∣ (⊤ ∧ ¬ 𝐵 = ∅)}
2016rabtru 3679 . . . . . . . . . 10 {𝑗𝐴 ∣ ⊤} = 𝐴
2120difeq1i 4117 . . . . . . . . 9 ({𝑗𝐴 ∣ ⊤} ∖ {𝑗𝐴𝐵 = ∅}) = (𝐴 ∖ {𝑗𝐴𝐵 = ∅})
22 truan 1550 . . . . . . . . . . 11 ((⊤ ∧ ¬ 𝐵 = ∅) ↔ ¬ 𝐵 = ∅)
23 df-ne 2939 . . . . . . . . . . 11 (𝐵 ≠ ∅ ↔ ¬ 𝐵 = ∅)
2422, 23bitr4i 277 . . . . . . . . . 10 ((⊤ ∧ ¬ 𝐵 = ∅) ↔ 𝐵 ≠ ∅)
2524rabbii 3436 . . . . . . . . 9 {𝑗𝐴 ∣ (⊤ ∧ ¬ 𝐵 = ∅)} = {𝑗𝐴𝐵 ≠ ∅}
2619, 21, 253eqtr3i 2766 . . . . . . . 8 (𝐴 ∖ {𝑗𝐴𝐵 = ∅}) = {𝑗𝐴𝐵 ≠ ∅}
2726a1i 11 . . . . . . 7 (𝜑 → (𝐴 ∖ {𝑗𝐴𝐵 = ∅}) = {𝑗𝐴𝐵 ≠ ∅})
28 eqidd 2731 . . . . . . 7 (𝜑𝐵 = 𝐵)
2915, 18, 9, 27, 28iuneq12df 5022 . . . . . 6 (𝜑 𝑗 ∈ (𝐴 ∖ {𝑗𝐴𝐵 = ∅})𝐵 = 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}𝐵)
30 rabid 3450 . . . . . . . . . . 11 (𝑗 ∈ {𝑗𝐴𝐵 = ∅} ↔ (𝑗𝐴𝐵 = ∅))
3130biimpi 215 . . . . . . . . . 10 (𝑗 ∈ {𝑗𝐴𝐵 = ∅} → (𝑗𝐴𝐵 = ∅))
3231adantl 480 . . . . . . . . 9 ((𝜑𝑗 ∈ {𝑗𝐴𝐵 = ∅}) → (𝑗𝐴𝐵 = ∅))
3332simprd 494 . . . . . . . 8 ((𝜑𝑗 ∈ {𝑗𝐴𝐵 = ∅}) → 𝐵 = ∅)
3433ralrimiva 3144 . . . . . . 7 (𝜑 → ∀𝑗 ∈ {𝑗𝐴𝐵 = ∅}𝐵 = ∅)
3517iunxdif3 5097 . . . . . . 7 (∀𝑗 ∈ {𝑗𝐴𝐵 = ∅}𝐵 = ∅ → 𝑗 ∈ (𝐴 ∖ {𝑗𝐴𝐵 = ∅})𝐵 = 𝑗𝐴 𝐵)
3634, 35syl 17 . . . . . 6 (𝜑 𝑗 ∈ (𝐴 ∖ {𝑗𝐴𝐵 = ∅})𝐵 = 𝑗𝐴 𝐵)
3729, 36eqtr3d 2772 . . . . 5 (𝜑 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}𝐵 = 𝑗𝐴 𝐵)
38 eqidd 2731 . . . . . . 7 (𝜑 → ({𝑗} × 𝐵) = ({𝑗} × 𝐵))
3915, 18, 9, 27, 38iuneq12df 5022 . . . . . 6 (𝜑 𝑗 ∈ (𝐴 ∖ {𝑗𝐴𝐵 = ∅})({𝑗} × 𝐵) = 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅} ({𝑗} × 𝐵))
4033xpeq2d 5705 . . . . . . . . 9 ((𝜑𝑗 ∈ {𝑗𝐴𝐵 = ∅}) → ({𝑗} × 𝐵) = ({𝑗} × ∅))
41 xp0 6156 . . . . . . . . 9 ({𝑗} × ∅) = ∅
4240, 41eqtrdi 2786 . . . . . . . 8 ((𝜑𝑗 ∈ {𝑗𝐴𝐵 = ∅}) → ({𝑗} × 𝐵) = ∅)
4342ralrimiva 3144 . . . . . . 7 (𝜑 → ∀𝑗 ∈ {𝑗𝐴𝐵 = ∅} ({𝑗} × 𝐵) = ∅)
4417iunxdif3 5097 . . . . . . 7 (∀𝑗 ∈ {𝑗𝐴𝐵 = ∅} ({𝑗} × 𝐵) = ∅ → 𝑗 ∈ (𝐴 ∖ {𝑗𝐴𝐵 = ∅})({𝑗} × 𝐵) = 𝑗𝐴 ({𝑗} × 𝐵))
4543, 44syl 17 . . . . . 6 (𝜑 𝑗 ∈ (𝐴 ∖ {𝑗𝐴𝐵 = ∅})({𝑗} × 𝐵) = 𝑗𝐴 ({𝑗} × 𝐵))
4639, 45eqtr3d 2772 . . . . 5 (𝜑 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅} ({𝑗} × 𝐵) = 𝑗𝐴 ({𝑗} × 𝐵))
4714, 37, 46f1eq123d 6824 . . . 4 (𝜑 → (𝑓: 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}𝐵1-1 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅} ({𝑗} × 𝐵) ↔ 𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵)))
4837raleqdv 3323 . . . 4 (𝜑 → (∀𝑘 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}𝐵(2nd ‘(𝑓𝑘)) = 𝑘 ↔ ∀𝑘 𝑗𝐴 𝐵(2nd ‘(𝑓𝑘)) = 𝑘))
4947, 48anbi12d 629 . . 3 (𝜑 → ((𝑓: 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}𝐵1-1 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅} ({𝑗} × 𝐵) ∧ ∀𝑘 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}𝐵(2nd ‘(𝑓𝑘)) = 𝑘) ↔ (𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵) ∧ ∀𝑘 𝑗𝐴 𝐵(2nd ‘(𝑓𝑘)) = 𝑘)))
5049exbidv 1922 . 2 (𝜑 → (∃𝑓(𝑓: 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}𝐵1-1 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅} ({𝑗} × 𝐵) ∧ ∀𝑘 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}𝐵(2nd ‘(𝑓𝑘)) = 𝑘) ↔ ∃𝑓(𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵) ∧ ∀𝑘 𝑗𝐴 𝐵(2nd ‘(𝑓𝑘)) = 𝑘)))
5113, 50mpbid 231 1 (𝜑 → ∃𝑓(𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵) ∧ ∀𝑘 𝑗𝐴 𝐵(2nd ‘(𝑓𝑘)) = 𝑘))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1539  wtru 1540  wex 1779  wcel 2104  wne 2938  wral 3059  {crab 3430  Vcvv 3472  cdif 3944  wss 3947  c0 4321  {csn 4627   ciun 4996   × cxp 5673  1-1wf1 6539  cfv 6542  2nd c2nd 7976
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-reg 9589  ax-inf2 9638  ax-ac2 10460
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7367  df-ov 7414  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-en 8942  df-r1 9761  df-rank 9762  df-card 9936  df-ac 10113
This theorem is referenced by:  fsumiunle  32302  esumiun  33390
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