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Theorem aciunf1 29693
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 3793 . . . 4 {𝑗𝐴𝐵 ≠ ∅} ⊆ 𝐴
2 aciunf1.0 . . . 4 (𝜑𝐴𝑉)
3 ssexg 4912 . . . 4 (({𝑗𝐴𝐵 ≠ ∅} ⊆ 𝐴𝐴𝑉) → {𝑗𝐴𝐵 ≠ ∅} ∈ V)
41, 2, 3sylancr 698 . . 3 (𝜑 → {𝑗𝐴𝐵 ≠ ∅} ∈ V)
5 rabid 3218 . . . . . 6 (𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅} ↔ (𝑗𝐴𝐵 ≠ ∅))
65biimpi 206 . . . . 5 (𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅} → (𝑗𝐴𝐵 ≠ ∅))
76adantl 473 . . . 4 ((𝜑𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}) → (𝑗𝐴𝐵 ≠ ∅))
87simprd 482 . . 3 ((𝜑𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}) → 𝐵 ≠ ∅)
9 nfrab1 3225 . . 3 𝑗{𝑗𝐴𝐵 ≠ ∅}
107simpld 477 . . . 4 ((𝜑𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}) → 𝑗𝐴)
11 aciunf1.1 . . . 4 ((𝜑𝑗𝐴) → 𝐵𝑊)
1210, 11syldan 488 . . 3 ((𝜑𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}) → 𝐵𝑊)
134, 8, 9, 12aciunf1lem 29692 . 2 (𝜑 → ∃𝑓(𝑓: 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}𝐵1-1 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅} ({𝑗} × 𝐵) ∧ ∀𝑘 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}𝐵(2nd ‘(𝑓𝑘)) = 𝑘))
14 eqidd 2725 . . . . 5 (𝜑𝑓 = 𝑓)
15 nfv 1956 . . . . . . 7 𝑗𝜑
16 nfcv 2866 . . . . . . . 8 𝑗𝐴
17 nfrab1 3225 . . . . . . . 8 𝑗{𝑗𝐴𝐵 = ∅}
1816, 17nfdif 3839 . . . . . . 7 𝑗(𝐴 ∖ {𝑗𝐴𝐵 = ∅})
19 difrab 4009 . . . . . . . . 9 ({𝑗𝐴 ∣ ⊤} ∖ {𝑗𝐴𝐵 = ∅}) = {𝑗𝐴 ∣ (⊤ ∧ ¬ 𝐵 = ∅)}
2016rabtru 3466 . . . . . . . . . 10 {𝑗𝐴 ∣ ⊤} = 𝐴
2120difeq1i 3832 . . . . . . . . 9 ({𝑗𝐴 ∣ ⊤} ∖ {𝑗𝐴𝐵 = ∅}) = (𝐴 ∖ {𝑗𝐴𝐵 = ∅})
22 truan 1614 . . . . . . . . . . 11 ((⊤ ∧ ¬ 𝐵 = ∅) ↔ ¬ 𝐵 = ∅)
23 df-ne 2897 . . . . . . . . . . 11 (𝐵 ≠ ∅ ↔ ¬ 𝐵 = ∅)
2422, 23bitr4i 267 . . . . . . . . . 10 ((⊤ ∧ ¬ 𝐵 = ∅) ↔ 𝐵 ≠ ∅)
2524rabbii 3289 . . . . . . . . 9 {𝑗𝐴 ∣ (⊤ ∧ ¬ 𝐵 = ∅)} = {𝑗𝐴𝐵 ≠ ∅}
2619, 21, 253eqtr3i 2754 . . . . . . . 8 (𝐴 ∖ {𝑗𝐴𝐵 = ∅}) = {𝑗𝐴𝐵 ≠ ∅}
2726a1i 11 . . . . . . 7 (𝜑 → (𝐴 ∖ {𝑗𝐴𝐵 = ∅}) = {𝑗𝐴𝐵 ≠ ∅})
28 eqidd 2725 . . . . . . 7 (𝜑𝐵 = 𝐵)
2915, 18, 9, 27, 28iuneq12df 4652 . . . . . 6 (𝜑 𝑗 ∈ (𝐴 ∖ {𝑗𝐴𝐵 = ∅})𝐵 = 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}𝐵)
30 rabid 3218 . . . . . . . . . . 11 (𝑗 ∈ {𝑗𝐴𝐵 = ∅} ↔ (𝑗𝐴𝐵 = ∅))
3130biimpi 206 . . . . . . . . . 10 (𝑗 ∈ {𝑗𝐴𝐵 = ∅} → (𝑗𝐴𝐵 = ∅))
3231adantl 473 . . . . . . . . 9 ((𝜑𝑗 ∈ {𝑗𝐴𝐵 = ∅}) → (𝑗𝐴𝐵 = ∅))
3332simprd 482 . . . . . . . 8 ((𝜑𝑗 ∈ {𝑗𝐴𝐵 = ∅}) → 𝐵 = ∅)
3433ralrimiva 3068 . . . . . . 7 (𝜑 → ∀𝑗 ∈ {𝑗𝐴𝐵 = ∅}𝐵 = ∅)
3517iunxdif3 4714 . . . . . . 7 (∀𝑗 ∈ {𝑗𝐴𝐵 = ∅}𝐵 = ∅ → 𝑗 ∈ (𝐴 ∖ {𝑗𝐴𝐵 = ∅})𝐵 = 𝑗𝐴 𝐵)
3634, 35syl 17 . . . . . 6 (𝜑 𝑗 ∈ (𝐴 ∖ {𝑗𝐴𝐵 = ∅})𝐵 = 𝑗𝐴 𝐵)
3729, 36eqtr3d 2760 . . . . 5 (𝜑 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}𝐵 = 𝑗𝐴 𝐵)
38 eqidd 2725 . . . . . . 7 (𝜑 → ({𝑗} × 𝐵) = ({𝑗} × 𝐵))
3915, 18, 9, 27, 38iuneq12df 4652 . . . . . 6 (𝜑 𝑗 ∈ (𝐴 ∖ {𝑗𝐴𝐵 = ∅})({𝑗} × 𝐵) = 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅} ({𝑗} × 𝐵))
4033xpeq2d 5248 . . . . . . . . 9 ((𝜑𝑗 ∈ {𝑗𝐴𝐵 = ∅}) → ({𝑗} × 𝐵) = ({𝑗} × ∅))
41 xp0 5662 . . . . . . . . 9 ({𝑗} × ∅) = ∅
4240, 41syl6eq 2774 . . . . . . . 8 ((𝜑𝑗 ∈ {𝑗𝐴𝐵 = ∅}) → ({𝑗} × 𝐵) = ∅)
4342ralrimiva 3068 . . . . . . 7 (𝜑 → ∀𝑗 ∈ {𝑗𝐴𝐵 = ∅} ({𝑗} × 𝐵) = ∅)
4417iunxdif3 4714 . . . . . . 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 2760 . . . . 5 (𝜑 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅} ({𝑗} × 𝐵) = 𝑗𝐴 ({𝑗} × 𝐵))
4714, 37, 46f1eq123d 6244 . . . 4 (𝜑 → (𝑓: 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}𝐵1-1 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅} ({𝑗} × 𝐵) ↔ 𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵)))
4837raleqdv 3247 . . . 4 (𝜑 → (∀𝑘 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}𝐵(2nd ‘(𝑓𝑘)) = 𝑘 ↔ ∀𝑘 𝑗𝐴 𝐵(2nd ‘(𝑓𝑘)) = 𝑘))
4947, 48anbi12d 749 . . 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 1963 . 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 1596  wtru 1597  wex 1817  wcel 2103  wne 2896  wral 3014  {crab 3018  Vcvv 3304  cdif 3677  wss 3680  c0 4023  {csn 4285   ciun 4628   × cxp 5216  1-1wf1 5998  cfv 6001  2nd c2nd 7284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-reg 8613  ax-inf2 8651  ax-ac2 9398
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-iin 4631  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-se 5178  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-isom 6010  df-riota 6726  df-om 7183  df-2nd 7286  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-en 8073  df-r1 8740  df-rank 8741  df-card 8878  df-ac 9052
This theorem is referenced by:  fsumiunle  29805  esumiun  30386
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