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Theorem aciunf1 29302
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 3666 . . . 4 {𝑗𝐴𝐵 ≠ ∅} ⊆ 𝐴
2 aciunf1.0 . . . 4 (𝜑𝐴𝑉)
3 ssexg 4764 . . . 4 (({𝑗𝐴𝐵 ≠ ∅} ⊆ 𝐴𝐴𝑉) → {𝑗𝐴𝐵 ≠ ∅} ∈ V)
41, 2, 3sylancr 694 . . 3 (𝜑 → {𝑗𝐴𝐵 ≠ ∅} ∈ V)
5 rabid 3106 . . . . . 6 (𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅} ↔ (𝑗𝐴𝐵 ≠ ∅))
65biimpi 206 . . . . 5 (𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅} → (𝑗𝐴𝐵 ≠ ∅))
76adantl 482 . . . 4 ((𝜑𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}) → (𝑗𝐴𝐵 ≠ ∅))
87simprd 479 . . 3 ((𝜑𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}) → 𝐵 ≠ ∅)
9 nfrab1 3111 . . 3 𝑗{𝑗𝐴𝐵 ≠ ∅}
107simpld 475 . . . 4 ((𝜑𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}) → 𝑗𝐴)
11 aciunf1.1 . . . 4 ((𝜑𝑗𝐴) → 𝐵𝑊)
1210, 11syldan 487 . . 3 ((𝜑𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}) → 𝐵𝑊)
134, 8, 9, 12aciunf1lem 29301 . 2 (𝜑 → ∃𝑓(𝑓: 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}𝐵1-1 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅} ({𝑗} × 𝐵) ∧ ∀𝑘 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}𝐵(2nd ‘(𝑓𝑘)) = 𝑘))
14 eqidd 2622 . . . . 5 (𝜑𝑓 = 𝑓)
15 nfv 1840 . . . . . . 7 𝑗𝜑
16 nfcv 2761 . . . . . . . 8 𝑗𝐴
17 nfrab1 3111 . . . . . . . 8 𝑗{𝑗𝐴𝐵 = ∅}
1816, 17nfdif 3709 . . . . . . 7 𝑗(𝐴 ∖ {𝑗𝐴𝐵 = ∅})
19 difrab 3877 . . . . . . . . 9 ({𝑗𝐴 ∣ ⊤} ∖ {𝑗𝐴𝐵 = ∅}) = {𝑗𝐴 ∣ (⊤ ∧ ¬ 𝐵 = ∅)}
2016rabtru 3344 . . . . . . . . . 10 {𝑗𝐴 ∣ ⊤} = 𝐴
2120difeq1i 3702 . . . . . . . . 9 ({𝑗𝐴 ∣ ⊤} ∖ {𝑗𝐴𝐵 = ∅}) = (𝐴 ∖ {𝑗𝐴𝐵 = ∅})
22 truan 1498 . . . . . . . . . . . . 13 ((⊤ ∧ ¬ 𝐵 = ∅) ↔ ¬ 𝐵 = ∅)
23 df-ne 2791 . . . . . . . . . . . . 13 (𝐵 ≠ ∅ ↔ ¬ 𝐵 = ∅)
2422, 23bitr4i 267 . . . . . . . . . . . 12 ((⊤ ∧ ¬ 𝐵 = ∅) ↔ 𝐵 ≠ ∅)
2524a1i 11 . . . . . . . . . . 11 (⊤ → ((⊤ ∧ ¬ 𝐵 = ∅) ↔ 𝐵 ≠ ∅))
2625rabbidv 3177 . . . . . . . . . 10 (⊤ → {𝑗𝐴 ∣ (⊤ ∧ ¬ 𝐵 = ∅)} = {𝑗𝐴𝐵 ≠ ∅})
2726trud 1490 . . . . . . . . 9 {𝑗𝐴 ∣ (⊤ ∧ ¬ 𝐵 = ∅)} = {𝑗𝐴𝐵 ≠ ∅}
2819, 21, 273eqtr3i 2651 . . . . . . . 8 (𝐴 ∖ {𝑗𝐴𝐵 = ∅}) = {𝑗𝐴𝐵 ≠ ∅}
2928a1i 11 . . . . . . 7 (𝜑 → (𝐴 ∖ {𝑗𝐴𝐵 = ∅}) = {𝑗𝐴𝐵 ≠ ∅})
30 eqidd 2622 . . . . . . 7 (𝜑𝐵 = 𝐵)
3115, 18, 9, 29, 30iuneq12df 4510 . . . . . 6 (𝜑 𝑗 ∈ (𝐴 ∖ {𝑗𝐴𝐵 = ∅})𝐵 = 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}𝐵)
32 rabid 3106 . . . . . . . . . . 11 (𝑗 ∈ {𝑗𝐴𝐵 = ∅} ↔ (𝑗𝐴𝐵 = ∅))
3332biimpi 206 . . . . . . . . . 10 (𝑗 ∈ {𝑗𝐴𝐵 = ∅} → (𝑗𝐴𝐵 = ∅))
3433adantl 482 . . . . . . . . 9 ((𝜑𝑗 ∈ {𝑗𝐴𝐵 = ∅}) → (𝑗𝐴𝐵 = ∅))
3534simprd 479 . . . . . . . 8 ((𝜑𝑗 ∈ {𝑗𝐴𝐵 = ∅}) → 𝐵 = ∅)
3635ralrimiva 2960 . . . . . . 7 (𝜑 → ∀𝑗 ∈ {𝑗𝐴𝐵 = ∅}𝐵 = ∅)
3717iunxdif3 4572 . . . . . . 7 (∀𝑗 ∈ {𝑗𝐴𝐵 = ∅}𝐵 = ∅ → 𝑗 ∈ (𝐴 ∖ {𝑗𝐴𝐵 = ∅})𝐵 = 𝑗𝐴 𝐵)
3836, 37syl 17 . . . . . 6 (𝜑 𝑗 ∈ (𝐴 ∖ {𝑗𝐴𝐵 = ∅})𝐵 = 𝑗𝐴 𝐵)
3931, 38eqtr3d 2657 . . . . 5 (𝜑 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}𝐵 = 𝑗𝐴 𝐵)
40 eqidd 2622 . . . . . . 7 (𝜑 → ({𝑗} × 𝐵) = ({𝑗} × 𝐵))
4115, 18, 9, 29, 40iuneq12df 4510 . . . . . 6 (𝜑 𝑗 ∈ (𝐴 ∖ {𝑗𝐴𝐵 = ∅})({𝑗} × 𝐵) = 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅} ({𝑗} × 𝐵))
4235xpeq2d 5099 . . . . . . . . 9 ((𝜑𝑗 ∈ {𝑗𝐴𝐵 = ∅}) → ({𝑗} × 𝐵) = ({𝑗} × ∅))
43 xp0 5511 . . . . . . . . 9 ({𝑗} × ∅) = ∅
4442, 43syl6eq 2671 . . . . . . . 8 ((𝜑𝑗 ∈ {𝑗𝐴𝐵 = ∅}) → ({𝑗} × 𝐵) = ∅)
4544ralrimiva 2960 . . . . . . 7 (𝜑 → ∀𝑗 ∈ {𝑗𝐴𝐵 = ∅} ({𝑗} × 𝐵) = ∅)
4617iunxdif3 4572 . . . . . . 7 (∀𝑗 ∈ {𝑗𝐴𝐵 = ∅} ({𝑗} × 𝐵) = ∅ → 𝑗 ∈ (𝐴 ∖ {𝑗𝐴𝐵 = ∅})({𝑗} × 𝐵) = 𝑗𝐴 ({𝑗} × 𝐵))
4745, 46syl 17 . . . . . 6 (𝜑 𝑗 ∈ (𝐴 ∖ {𝑗𝐴𝐵 = ∅})({𝑗} × 𝐵) = 𝑗𝐴 ({𝑗} × 𝐵))
4841, 47eqtr3d 2657 . . . . 5 (𝜑 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅} ({𝑗} × 𝐵) = 𝑗𝐴 ({𝑗} × 𝐵))
4914, 39, 48f1eq123d 6088 . . . 4 (𝜑 → (𝑓: 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}𝐵1-1 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅} ({𝑗} × 𝐵) ↔ 𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵)))
5039raleqdv 3133 . . . 4 (𝜑 → (∀𝑘 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}𝐵(2nd ‘(𝑓𝑘)) = 𝑘 ↔ ∀𝑘 𝑗𝐴 𝐵(2nd ‘(𝑓𝑘)) = 𝑘))
5149, 50anbi12d 746 . . 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)))
5251exbidv 1847 . 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)))
5313, 52mpbid 222 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  wb 196  wa 384   = wceq 1480  wtru 1481  wex 1701  wcel 1987  wne 2790  wral 2907  {crab 2911  Vcvv 3186  cdif 3552  wss 3555  c0 3891  {csn 4148   ciun 4485   × cxp 5072  1-1wf1 5844  cfv 5847  2nd c2nd 7112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-reg 8441  ax-inf2 8482  ax-ac2 9229
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-om 7013  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-en 7900  df-r1 8571  df-rank 8572  df-card 8709  df-ac 8883
This theorem is referenced by:  esumiun  29934
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