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Theorem aciunf1 31579
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 4037 . . . 4 {𝑗𝐴𝐵 ≠ ∅} ⊆ 𝐴
2 aciunf1.0 . . . 4 (𝜑𝐴𝑉)
3 ssexg 5280 . . . 4 (({𝑗𝐴𝐵 ≠ ∅} ⊆ 𝐴𝐴𝑉) → {𝑗𝐴𝐵 ≠ ∅} ∈ V)
41, 2, 3sylancr 587 . . 3 (𝜑 → {𝑗𝐴𝐵 ≠ ∅} ∈ V)
5 rabid 3427 . . . . . 6 (𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅} ↔ (𝑗𝐴𝐵 ≠ ∅))
65biimpi 215 . . . . 5 (𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅} → (𝑗𝐴𝐵 ≠ ∅))
76adantl 482 . . . 4 ((𝜑𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}) → (𝑗𝐴𝐵 ≠ ∅))
87simprd 496 . . 3 ((𝜑𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}) → 𝐵 ≠ ∅)
9 nfrab1 3426 . . 3 𝑗{𝑗𝐴𝐵 ≠ ∅}
107simpld 495 . . . 4 ((𝜑𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}) → 𝑗𝐴)
11 aciunf1.1 . . . 4 ((𝜑𝑗𝐴) → 𝐵𝑊)
1210, 11syldan 591 . . 3 ((𝜑𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}) → 𝐵𝑊)
134, 8, 9, 12aciunf1lem 31578 . 2 (𝜑 → ∃𝑓(𝑓: 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}𝐵1-1 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅} ({𝑗} × 𝐵) ∧ ∀𝑘 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}𝐵(2nd ‘(𝑓𝑘)) = 𝑘))
14 eqidd 2737 . . . . 5 (𝜑𝑓 = 𝑓)
15 nfv 1917 . . . . . . 7 𝑗𝜑
16 nfcv 2907 . . . . . . . 8 𝑗𝐴
17 nfrab1 3426 . . . . . . . 8 𝑗{𝑗𝐴𝐵 = ∅}
1816, 17nfdif 4085 . . . . . . 7 𝑗(𝐴 ∖ {𝑗𝐴𝐵 = ∅})
19 difrab 4268 . . . . . . . . 9 ({𝑗𝐴 ∣ ⊤} ∖ {𝑗𝐴𝐵 = ∅}) = {𝑗𝐴 ∣ (⊤ ∧ ¬ 𝐵 = ∅)}
2016rabtru 3642 . . . . . . . . . 10 {𝑗𝐴 ∣ ⊤} = 𝐴
2120difeq1i 4078 . . . . . . . . 9 ({𝑗𝐴 ∣ ⊤} ∖ {𝑗𝐴𝐵 = ∅}) = (𝐴 ∖ {𝑗𝐴𝐵 = ∅})
22 truan 1552 . . . . . . . . . . 11 ((⊤ ∧ ¬ 𝐵 = ∅) ↔ ¬ 𝐵 = ∅)
23 df-ne 2944 . . . . . . . . . . 11 (𝐵 ≠ ∅ ↔ ¬ 𝐵 = ∅)
2422, 23bitr4i 277 . . . . . . . . . 10 ((⊤ ∧ ¬ 𝐵 = ∅) ↔ 𝐵 ≠ ∅)
2524rabbii 3413 . . . . . . . . 9 {𝑗𝐴 ∣ (⊤ ∧ ¬ 𝐵 = ∅)} = {𝑗𝐴𝐵 ≠ ∅}
2619, 21, 253eqtr3i 2772 . . . . . . . 8 (𝐴 ∖ {𝑗𝐴𝐵 = ∅}) = {𝑗𝐴𝐵 ≠ ∅}
2726a1i 11 . . . . . . 7 (𝜑 → (𝐴 ∖ {𝑗𝐴𝐵 = ∅}) = {𝑗𝐴𝐵 ≠ ∅})
28 eqidd 2737 . . . . . . 7 (𝜑𝐵 = 𝐵)
2915, 18, 9, 27, 28iuneq12df 4980 . . . . . 6 (𝜑 𝑗 ∈ (𝐴 ∖ {𝑗𝐴𝐵 = ∅})𝐵 = 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}𝐵)
30 rabid 3427 . . . . . . . . . . 11 (𝑗 ∈ {𝑗𝐴𝐵 = ∅} ↔ (𝑗𝐴𝐵 = ∅))
3130biimpi 215 . . . . . . . . . 10 (𝑗 ∈ {𝑗𝐴𝐵 = ∅} → (𝑗𝐴𝐵 = ∅))
3231adantl 482 . . . . . . . . 9 ((𝜑𝑗 ∈ {𝑗𝐴𝐵 = ∅}) → (𝑗𝐴𝐵 = ∅))
3332simprd 496 . . . . . . . 8 ((𝜑𝑗 ∈ {𝑗𝐴𝐵 = ∅}) → 𝐵 = ∅)
3433ralrimiva 3143 . . . . . . 7 (𝜑 → ∀𝑗 ∈ {𝑗𝐴𝐵 = ∅}𝐵 = ∅)
3517iunxdif3 5055 . . . . . . 7 (∀𝑗 ∈ {𝑗𝐴𝐵 = ∅}𝐵 = ∅ → 𝑗 ∈ (𝐴 ∖ {𝑗𝐴𝐵 = ∅})𝐵 = 𝑗𝐴 𝐵)
3634, 35syl 17 . . . . . 6 (𝜑 𝑗 ∈ (𝐴 ∖ {𝑗𝐴𝐵 = ∅})𝐵 = 𝑗𝐴 𝐵)
3729, 36eqtr3d 2778 . . . . 5 (𝜑 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}𝐵 = 𝑗𝐴 𝐵)
38 eqidd 2737 . . . . . . 7 (𝜑 → ({𝑗} × 𝐵) = ({𝑗} × 𝐵))
3915, 18, 9, 27, 38iuneq12df 4980 . . . . . 6 (𝜑 𝑗 ∈ (𝐴 ∖ {𝑗𝐴𝐵 = ∅})({𝑗} × 𝐵) = 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅} ({𝑗} × 𝐵))
4033xpeq2d 5663 . . . . . . . . 9 ((𝜑𝑗 ∈ {𝑗𝐴𝐵 = ∅}) → ({𝑗} × 𝐵) = ({𝑗} × ∅))
41 xp0 6110 . . . . . . . . 9 ({𝑗} × ∅) = ∅
4240, 41eqtrdi 2792 . . . . . . . 8 ((𝜑𝑗 ∈ {𝑗𝐴𝐵 = ∅}) → ({𝑗} × 𝐵) = ∅)
4342ralrimiva 3143 . . . . . . 7 (𝜑 → ∀𝑗 ∈ {𝑗𝐴𝐵 = ∅} ({𝑗} × 𝐵) = ∅)
4417iunxdif3 5055 . . . . . . 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 2778 . . . . 5 (𝜑 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅} ({𝑗} × 𝐵) = 𝑗𝐴 ({𝑗} × 𝐵))
4714, 37, 46f1eq123d 6776 . . . 4 (𝜑 → (𝑓: 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}𝐵1-1 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅} ({𝑗} × 𝐵) ↔ 𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵)))
4837raleqdv 3313 . . . 4 (𝜑 → (∀𝑘 𝑗 ∈ {𝑗𝐴𝐵 ≠ ∅}𝐵(2nd ‘(𝑓𝑘)) = 𝑘 ↔ ∀𝑘 𝑗𝐴 𝐵(2nd ‘(𝑓𝑘)) = 𝑘))
4947, 48anbi12d 631 . . 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 1924 . 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 396   = wceq 1541  wtru 1542  wex 1781  wcel 2106  wne 2943  wral 3064  {crab 3407  Vcvv 3445  cdif 3907  wss 3910  c0 4282  {csn 4586   ciun 4954   × cxp 5631  1-1wf1 6493  cfv 6496  2nd c2nd 7920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-reg 9528  ax-inf2 9577  ax-ac2 10399
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-en 8884  df-r1 9700  df-rank 9701  df-card 9875  df-ac 10052
This theorem is referenced by:  fsumiunle  31725  esumiun  32693
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