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Theorem aciunf1 29591
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 3720 . . . 4 {𝑗𝐴𝐵 ≠ ∅} ⊆ 𝐴
2 aciunf1.0 . . . 4 (𝜑𝐴𝑉)
3 ssexg 4837 . . . 4 (({𝑗𝐴𝐵 ≠ ∅} ⊆ 𝐴𝐴𝑉) → {𝑗𝐴𝐵 ≠ ∅} ∈ V)
41, 2, 3sylancr 696 . . 3 (𝜑 → {𝑗𝐴𝐵 ≠ ∅} ∈ V)
5 rabid 3145 . . . . . 6 (𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅} ↔ (𝑗𝐴𝐵 ≠ ∅))
65biimpi 206 . . . . 5 (𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅} → (𝑗𝐴𝐵 ≠ ∅))
76adantl 481 . . . 4 ((𝜑𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}) → (𝑗𝐴𝐵 ≠ ∅))
87simprd 478 . . 3 ((𝜑𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}) → 𝐵 ≠ ∅)
9 nfrab1 3152 . . 3 𝑗{𝑗𝐴𝐵 ≠ ∅}
107simpld 474 . . . 4 ((𝜑𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}) → 𝑗𝐴)
11 aciunf1.1 . . . 4 ((𝜑𝑗𝐴) → 𝐵𝑊)
1210, 11syldan 486 . . 3 ((𝜑𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}) → 𝐵𝑊)
134, 8, 9, 12aciunf1lem 29590 . 2 (𝜑 → ∃𝑓(𝑓: 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}𝐵1-1 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅} ({𝑗} × 𝐵) ∧ ∀𝑘 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}𝐵(2nd ‘(𝑓𝑘)) = 𝑘))
14 eqidd 2652 . . . . 5 (𝜑𝑓 = 𝑓)
15 nfv 1883 . . . . . . 7 𝑗𝜑
16 nfcv 2793 . . . . . . . 8 𝑗𝐴
17 nfrab1 3152 . . . . . . . 8 𝑗{𝑗𝐴𝐵 = ∅}
1816, 17nfdif 3764 . . . . . . 7 𝑗(𝐴 ∖ {𝑗𝐴𝐵 = ∅})
19 difrab 3934 . . . . . . . . 9 ({𝑗𝐴 ∣ ⊤} ∖ {𝑗𝐴𝐵 = ∅}) = {𝑗𝐴 ∣ (⊤ ∧ ¬ 𝐵 = ∅)}
2016rabtru 3393 . . . . . . . . . 10 {𝑗𝐴 ∣ ⊤} = 𝐴
2120difeq1i 3757 . . . . . . . . 9 ({𝑗𝐴 ∣ ⊤} ∖ {𝑗𝐴𝐵 = ∅}) = (𝐴 ∖ {𝑗𝐴𝐵 = ∅})
22 truan 1541 . . . . . . . . . . 11 ((⊤ ∧ ¬ 𝐵 = ∅) ↔ ¬ 𝐵 = ∅)
23 df-ne 2824 . . . . . . . . . . 11 (𝐵 ≠ ∅ ↔ ¬ 𝐵 = ∅)
2422, 23bitr4i 267 . . . . . . . . . 10 ((⊤ ∧ ¬ 𝐵 = ∅) ↔ 𝐵 ≠ ∅)
2524rabbii 3216 . . . . . . . . 9 {𝑗𝐴 ∣ (⊤ ∧ ¬ 𝐵 = ∅)} = {𝑗𝐴𝐵 ≠ ∅}
2619, 21, 253eqtr3i 2681 . . . . . . . 8 (𝐴 ∖ {𝑗𝐴𝐵 = ∅}) = {𝑗𝐴𝐵 ≠ ∅}
2726a1i 11 . . . . . . 7 (𝜑 → (𝐴 ∖ {𝑗𝐴𝐵 = ∅}) = {𝑗𝐴𝐵 ≠ ∅})
28 eqidd 2652 . . . . . . 7 (𝜑𝐵 = 𝐵)
2915, 18, 9, 27, 28iuneq12df 4576 . . . . . 6 (𝜑 𝑗 ∈ (𝐴 ∖ {𝑗𝐴𝐵 = ∅})𝐵 = 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}𝐵)
30 rabid 3145 . . . . . . . . . . 11 (𝑗 ∈ {𝑗𝐴𝐵 = ∅} ↔ (𝑗𝐴𝐵 = ∅))
3130biimpi 206 . . . . . . . . . 10 (𝑗 ∈ {𝑗𝐴𝐵 = ∅} → (𝑗𝐴𝐵 = ∅))
3231adantl 481 . . . . . . . . 9 ((𝜑𝑗 ∈ {𝑗𝐴𝐵 = ∅}) → (𝑗𝐴𝐵 = ∅))
3332simprd 478 . . . . . . . 8 ((𝜑𝑗 ∈ {𝑗𝐴𝐵 = ∅}) → 𝐵 = ∅)
3433ralrimiva 2995 . . . . . . 7 (𝜑 → ∀𝑗 ∈ {𝑗𝐴𝐵 = ∅}𝐵 = ∅)
3517iunxdif3 4638 . . . . . . 7 (∀𝑗 ∈ {𝑗𝐴𝐵 = ∅}𝐵 = ∅ → 𝑗 ∈ (𝐴 ∖ {𝑗𝐴𝐵 = ∅})𝐵 = 𝑗𝐴 𝐵)
3634, 35syl 17 . . . . . 6 (𝜑 𝑗 ∈ (𝐴 ∖ {𝑗𝐴𝐵 = ∅})𝐵 = 𝑗𝐴 𝐵)
3729, 36eqtr3d 2687 . . . . 5 (𝜑 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}𝐵 = 𝑗𝐴 𝐵)
38 eqidd 2652 . . . . . . 7 (𝜑 → ({𝑗} × 𝐵) = ({𝑗} × 𝐵))
3915, 18, 9, 27, 38iuneq12df 4576 . . . . . 6 (𝜑 𝑗 ∈ (𝐴 ∖ {𝑗𝐴𝐵 = ∅})({𝑗} × 𝐵) = 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅} ({𝑗} × 𝐵))
4033xpeq2d 5173 . . . . . . . . 9 ((𝜑𝑗 ∈ {𝑗𝐴𝐵 = ∅}) → ({𝑗} × 𝐵) = ({𝑗} × ∅))
41 xp0 5587 . . . . . . . . 9 ({𝑗} × ∅) = ∅
4240, 41syl6eq 2701 . . . . . . . 8 ((𝜑𝑗 ∈ {𝑗𝐴𝐵 = ∅}) → ({𝑗} × 𝐵) = ∅)
4342ralrimiva 2995 . . . . . . 7 (𝜑 → ∀𝑗 ∈ {𝑗𝐴𝐵 = ∅} ({𝑗} × 𝐵) = ∅)
4417iunxdif3 4638 . . . . . . 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 2687 . . . . 5 (𝜑 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅} ({𝑗} × 𝐵) = 𝑗𝐴 ({𝑗} × 𝐵))
4714, 37, 46f1eq123d 6169 . . . 4 (𝜑 → (𝑓: 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}𝐵1-1 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅} ({𝑗} × 𝐵) ↔ 𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵)))
4837raleqdv 3174 . . . 4 (𝜑 → (∀𝑘 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}𝐵(2nd ‘(𝑓𝑘)) = 𝑘 ↔ ∀𝑘 𝑗𝐴 𝐵(2nd ‘(𝑓𝑘)) = 𝑘))
4947, 48anbi12d 747 . . 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 1890 . 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 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  wa 383   = wceq 1523  wtru 1524  wex 1744  wcel 2030  wne 2823  wral 2941  {crab 2945  Vcvv 3231  cdif 3604  wss 3607  c0 3948  {csn 4210   ciun 4552   × cxp 5141  1-1wf1 5923  cfv 5926  2nd c2nd 7209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-reg 8538  ax-inf2 8576  ax-ac2 9323
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-om 7108  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-en 7998  df-r1 8665  df-rank 8666  df-card 8803  df-ac 8977
This theorem is referenced by:  fsumiunle  29703  esumiun  30284
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