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Theorem acunirnmpt2f 30414
Description: Axiom of choice for the union of the range of a mapping to function. (Contributed by Thierry Arnoux, 7-Nov-2019.)
Hypotheses
Ref Expression
acunirnmpt.0 (𝜑𝐴𝑉)
acunirnmpt.1 ((𝜑𝑗𝐴) → 𝐵 ≠ ∅)
aciunf1lem.a 𝑗𝐴
acunirnmpt2f.c 𝑗𝐶
acunirnmpt2f.d 𝑗𝐷
acunirnmpt2f.2 𝐶 = 𝑗𝐴 𝐵
acunirnmpt2f.3 (𝑗 = (𝑓𝑥) → 𝐵 = 𝐷)
acunirnmpt2f.4 ((𝜑𝑗𝐴) → 𝐵𝑊)
Assertion
Ref Expression
acunirnmpt2f (𝜑 → ∃𝑓(𝑓:𝐶𝐴 ∧ ∀𝑥𝐶 𝑥𝐷))
Distinct variable groups:   𝑥,𝑓,𝐴   𝐵,𝑓   𝐶,𝑓,𝑥   𝑓,𝑗,𝜑,𝑥
Allowed substitution hints:   𝐴(𝑗)   𝐵(𝑥,𝑗)   𝐶(𝑗)   𝐷(𝑥,𝑓,𝑗)   𝑉(𝑥,𝑓,𝑗)   𝑊(𝑥,𝑓,𝑗)

Proof of Theorem acunirnmpt2f
Dummy variables 𝑐 𝑦 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplr 768 . . . . . 6 ((((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦) → 𝑦 ∈ ran (𝑗𝐴𝐵))
2 vex 3472 . . . . . . 7 𝑦 ∈ V
3 eqid 2822 . . . . . . . 8 (𝑗𝐴𝐵) = (𝑗𝐴𝐵)
43elrnmpt 5805 . . . . . . 7 (𝑦 ∈ V → (𝑦 ∈ ran (𝑗𝐴𝐵) ↔ ∃𝑗𝐴 𝑦 = 𝐵))
52, 4ax-mp 5 . . . . . 6 (𝑦 ∈ ran (𝑗𝐴𝐵) ↔ ∃𝑗𝐴 𝑦 = 𝐵)
61, 5sylib 221 . . . . 5 ((((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦) → ∃𝑗𝐴 𝑦 = 𝐵)
7 nfv 1915 . . . . . . . . 9 𝑗𝜑
8 acunirnmpt2f.c . . . . . . . . . 10 𝑗𝐶
98nfcri 2967 . . . . . . . . 9 𝑗 𝑥𝐶
107, 9nfan 1900 . . . . . . . 8 𝑗(𝜑𝑥𝐶)
11 nfcv 2979 . . . . . . . . 9 𝑗𝑦
12 nfmpt1 5140 . . . . . . . . . 10 𝑗(𝑗𝐴𝐵)
1312nfrn 5801 . . . . . . . . 9 𝑗ran (𝑗𝐴𝐵)
1411, 13nfel 2993 . . . . . . . 8 𝑗 𝑦 ∈ ran (𝑗𝐴𝐵)
1510, 14nfan 1900 . . . . . . 7 𝑗((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵))
16 nfv 1915 . . . . . . 7 𝑗 𝑥𝑦
1715, 16nfan 1900 . . . . . 6 𝑗(((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦)
18 simpllr 775 . . . . . . . . 9 ((((((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → 𝑥𝑦)
19 simpr 488 . . . . . . . . 9 ((((((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
2018, 19eleqtrd 2916 . . . . . . . 8 ((((((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → 𝑥𝐵)
2120ex 416 . . . . . . 7 (((((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦) ∧ 𝑗𝐴) → (𝑦 = 𝐵𝑥𝐵))
2221ex 416 . . . . . 6 ((((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦) → (𝑗𝐴 → (𝑦 = 𝐵𝑥𝐵)))
2317, 22reximdai 3297 . . . . 5 ((((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦) → (∃𝑗𝐴 𝑦 = 𝐵 → ∃𝑗𝐴 𝑥𝐵))
246, 23mpd 15 . . . 4 ((((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦) → ∃𝑗𝐴 𝑥𝐵)
25 acunirnmpt2f.2 . . . . . . . 8 𝐶 = 𝑗𝐴 𝐵
26 acunirnmpt2f.4 . . . . . . . . . 10 ((𝜑𝑗𝐴) → 𝐵𝑊)
2726ralrimiva 3174 . . . . . . . . 9 (𝜑 → ∀𝑗𝐴 𝐵𝑊)
28 dfiun3g 5813 . . . . . . . . 9 (∀𝑗𝐴 𝐵𝑊 𝑗𝐴 𝐵 = ran (𝑗𝐴𝐵))
2927, 28syl 17 . . . . . . . 8 (𝜑 𝑗𝐴 𝐵 = ran (𝑗𝐴𝐵))
3025, 29syl5eq 2869 . . . . . . 7 (𝜑𝐶 = ran (𝑗𝐴𝐵))
3130eleq2d 2899 . . . . . 6 (𝜑 → (𝑥𝐶𝑥 ran (𝑗𝐴𝐵)))
3231biimpa 480 . . . . 5 ((𝜑𝑥𝐶) → 𝑥 ran (𝑗𝐴𝐵))
33 eluni2 4817 . . . . 5 (𝑥 ran (𝑗𝐴𝐵) ↔ ∃𝑦 ∈ ran (𝑗𝐴𝐵)𝑥𝑦)
3432, 33sylib 221 . . . 4 ((𝜑𝑥𝐶) → ∃𝑦 ∈ ran (𝑗𝐴𝐵)𝑥𝑦)
3524, 34r19.29a 3275 . . 3 ((𝜑𝑥𝐶) → ∃𝑗𝐴 𝑥𝐵)
3635ralrimiva 3174 . 2 (𝜑 → ∀𝑥𝐶𝑗𝐴 𝑥𝐵)
37 acunirnmpt.0 . . . . 5 (𝜑𝐴𝑉)
38 aciunf1lem.a . . . . . . 7 𝑗𝐴
39 nfcv 2979 . . . . . . 7 𝑘𝐴
40 nfcv 2979 . . . . . . 7 𝑘𝐵
41 nfcsb1v 3879 . . . . . . 7 𝑗𝑘 / 𝑗𝐵
42 csbeq1a 3869 . . . . . . 7 (𝑗 = 𝑘𝐵 = 𝑘 / 𝑗𝐵)
4338, 39, 40, 41, 42cbvmptf 5141 . . . . . 6 (𝑗𝐴𝐵) = (𝑘𝐴𝑘 / 𝑗𝐵)
44 mptexg 6966 . . . . . 6 (𝐴𝑉 → (𝑘𝐴𝑘 / 𝑗𝐵) ∈ V)
4543, 44eqeltrid 2918 . . . . 5 (𝐴𝑉 → (𝑗𝐴𝐵) ∈ V)
46 rnexg 7600 . . . . 5 ((𝑗𝐴𝐵) ∈ V → ran (𝑗𝐴𝐵) ∈ V)
47 uniexg 7451 . . . . 5 (ran (𝑗𝐴𝐵) ∈ V → ran (𝑗𝐴𝐵) ∈ V)
4837, 45, 46, 474syl 19 . . . 4 (𝜑 ran (𝑗𝐴𝐵) ∈ V)
4930, 48eqeltrd 2914 . . 3 (𝜑𝐶 ∈ V)
50 id 22 . . . . . 6 (𝑐 = 𝐶𝑐 = 𝐶)
5150raleqdv 3392 . . . . 5 (𝑐 = 𝐶 → (∀𝑥𝑐𝑗𝐴 𝑥𝐵 ↔ ∀𝑥𝐶𝑗𝐴 𝑥𝐵))
5250feq2d 6480 . . . . . . 7 (𝑐 = 𝐶 → (𝑓:𝑐𝐴𝑓:𝐶𝐴))
5350raleqdv 3392 . . . . . . 7 (𝑐 = 𝐶 → (∀𝑥𝑐 𝑥𝐷 ↔ ∀𝑥𝐶 𝑥𝐷))
5452, 53anbi12d 633 . . . . . 6 (𝑐 = 𝐶 → ((𝑓:𝑐𝐴 ∧ ∀𝑥𝑐 𝑥𝐷) ↔ (𝑓:𝐶𝐴 ∧ ∀𝑥𝐶 𝑥𝐷)))
5554exbidv 1922 . . . . 5 (𝑐 = 𝐶 → (∃𝑓(𝑓:𝑐𝐴 ∧ ∀𝑥𝑐 𝑥𝐷) ↔ ∃𝑓(𝑓:𝐶𝐴 ∧ ∀𝑥𝐶 𝑥𝐷)))
5651, 55imbi12d 348 . . . 4 (𝑐 = 𝐶 → ((∀𝑥𝑐𝑗𝐴 𝑥𝐵 → ∃𝑓(𝑓:𝑐𝐴 ∧ ∀𝑥𝑐 𝑥𝐷)) ↔ (∀𝑥𝐶𝑗𝐴 𝑥𝐵 → ∃𝑓(𝑓:𝐶𝐴 ∧ ∀𝑥𝐶 𝑥𝐷))))
57 acunirnmpt2f.d . . . . . 6 𝑗𝐷
5857nfcri 2967 . . . . 5 𝑗 𝑥𝐷
59 vex 3472 . . . . 5 𝑐 ∈ V
60 acunirnmpt2f.3 . . . . . 6 (𝑗 = (𝑓𝑥) → 𝐵 = 𝐷)
6160eleq2d 2899 . . . . 5 (𝑗 = (𝑓𝑥) → (𝑥𝐵𝑥𝐷))
6238, 58, 59, 61ac6sf2 30378 . . . 4 (∀𝑥𝑐𝑗𝐴 𝑥𝐵 → ∃𝑓(𝑓:𝑐𝐴 ∧ ∀𝑥𝑐 𝑥𝐷))
6356, 62vtoclg 3542 . . 3 (𝐶 ∈ V → (∀𝑥𝐶𝑗𝐴 𝑥𝐵 → ∃𝑓(𝑓:𝐶𝐴 ∧ ∀𝑥𝐶 𝑥𝐷)))
6449, 63syl 17 . 2 (𝜑 → (∀𝑥𝐶𝑗𝐴 𝑥𝐵 → ∃𝑓(𝑓:𝐶𝐴 ∧ ∀𝑥𝐶 𝑥𝐷)))
6536, 64mpd 15 1 (𝜑 → ∃𝑓(𝑓:𝐶𝐴 ∧ ∀𝑥𝐶 𝑥𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wex 1781  wcel 2114  wnfc 2960  wne 3011  wral 3130  wrex 3131  Vcvv 3469  csb 3855  c0 4265   cuni 4813   ciun 4894  cmpt 5122  ran crn 5533  wf 6330  cfv 6334
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-reg 9044  ax-inf2 9092  ax-ac2 9874
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-iin 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-se 5492  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-isom 6343  df-riota 7098  df-om 7566  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-en 8497  df-r1 9181  df-rank 9182  df-card 9356  df-ac 9531
This theorem is referenced by:  aciunf1lem  30415
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