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Theorem acunirnmpt2f 32671
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 769 . . . . . 6 ((((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦) → 𝑦 ∈ ran (𝑗𝐴𝐵))
2 vex 3484 . . . . . . 7 𝑦 ∈ V
3 eqid 2737 . . . . . . . 8 (𝑗𝐴𝐵) = (𝑗𝐴𝐵)
43elrnmpt 5969 . . . . . . 7 (𝑦 ∈ V → (𝑦 ∈ ran (𝑗𝐴𝐵) ↔ ∃𝑗𝐴 𝑦 = 𝐵))
52, 4ax-mp 5 . . . . . 6 (𝑦 ∈ ran (𝑗𝐴𝐵) ↔ ∃𝑗𝐴 𝑦 = 𝐵)
61, 5sylib 218 . . . . 5 ((((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦) → ∃𝑗𝐴 𝑦 = 𝐵)
7 nfv 1914 . . . . . . . . 9 𝑗𝜑
8 acunirnmpt2f.c . . . . . . . . . 10 𝑗𝐶
98nfcri 2897 . . . . . . . . 9 𝑗 𝑥𝐶
107, 9nfan 1899 . . . . . . . 8 𝑗(𝜑𝑥𝐶)
11 nfcv 2905 . . . . . . . . 9 𝑗𝑦
12 nfmpt1 5250 . . . . . . . . . 10 𝑗(𝑗𝐴𝐵)
1312nfrn 5963 . . . . . . . . 9 𝑗ran (𝑗𝐴𝐵)
1411, 13nfel 2920 . . . . . . . 8 𝑗 𝑦 ∈ ran (𝑗𝐴𝐵)
1510, 14nfan 1899 . . . . . . 7 𝑗((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵))
16 nfv 1914 . . . . . . 7 𝑗 𝑥𝑦
1715, 16nfan 1899 . . . . . 6 𝑗(((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦)
18 simpllr 776 . . . . . . . . 9 ((((((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → 𝑥𝑦)
19 simpr 484 . . . . . . . . 9 ((((((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
2018, 19eleqtrd 2843 . . . . . . . 8 ((((((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → 𝑥𝐵)
2120ex 412 . . . . . . 7 (((((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦) ∧ 𝑗𝐴) → (𝑦 = 𝐵𝑥𝐵))
2221ex 412 . . . . . 6 ((((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦) → (𝑗𝐴 → (𝑦 = 𝐵𝑥𝐵)))
2317, 22reximdai 3261 . . . . 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 3146 . . . . . . . . 9 (𝜑 → ∀𝑗𝐴 𝐵𝑊)
28 dfiun3g 5978 . . . . . . . . 9 (∀𝑗𝐴 𝐵𝑊 𝑗𝐴 𝐵 = ran (𝑗𝐴𝐵))
2927, 28syl 17 . . . . . . . 8 (𝜑 𝑗𝐴 𝐵 = ran (𝑗𝐴𝐵))
3025, 29eqtrid 2789 . . . . . . 7 (𝜑𝐶 = ran (𝑗𝐴𝐵))
3130eleq2d 2827 . . . . . 6 (𝜑 → (𝑥𝐶𝑥 ran (𝑗𝐴𝐵)))
3231biimpa 476 . . . . 5 ((𝜑𝑥𝐶) → 𝑥 ran (𝑗𝐴𝐵))
33 eluni2 4911 . . . . 5 (𝑥 ran (𝑗𝐴𝐵) ↔ ∃𝑦 ∈ ran (𝑗𝐴𝐵)𝑥𝑦)
3432, 33sylib 218 . . . 4 ((𝜑𝑥𝐶) → ∃𝑦 ∈ ran (𝑗𝐴𝐵)𝑥𝑦)
3524, 34r19.29a 3162 . . 3 ((𝜑𝑥𝐶) → ∃𝑗𝐴 𝑥𝐵)
3635ralrimiva 3146 . 2 (𝜑 → ∀𝑥𝐶𝑗𝐴 𝑥𝐵)
37 acunirnmpt.0 . . . . 5 (𝜑𝐴𝑉)
38 aciunf1lem.a . . . . . . 7 𝑗𝐴
39 nfcv 2905 . . . . . . 7 𝑘𝐴
40 nfcv 2905 . . . . . . 7 𝑘𝐵
41 nfcsb1v 3923 . . . . . . 7 𝑗𝑘 / 𝑗𝐵
42 csbeq1a 3913 . . . . . . 7 (𝑗 = 𝑘𝐵 = 𝑘 / 𝑗𝐵)
4338, 39, 40, 41, 42cbvmptf 5251 . . . . . 6 (𝑗𝐴𝐵) = (𝑘𝐴𝑘 / 𝑗𝐵)
44 mptexg 7241 . . . . . 6 (𝐴𝑉 → (𝑘𝐴𝑘 / 𝑗𝐵) ∈ V)
4543, 44eqeltrid 2845 . . . . 5 (𝐴𝑉 → (𝑗𝐴𝐵) ∈ V)
46 rnexg 7924 . . . . 5 ((𝑗𝐴𝐵) ∈ V → ran (𝑗𝐴𝐵) ∈ V)
47 uniexg 7760 . . . . 5 (ran (𝑗𝐴𝐵) ∈ V → ran (𝑗𝐴𝐵) ∈ V)
4837, 45, 46, 474syl 19 . . . 4 (𝜑 ran (𝑗𝐴𝐵) ∈ V)
4930, 48eqeltrd 2841 . . 3 (𝜑𝐶 ∈ V)
50 id 22 . . . . . 6 (𝑐 = 𝐶𝑐 = 𝐶)
5150raleqdv 3326 . . . . 5 (𝑐 = 𝐶 → (∀𝑥𝑐𝑗𝐴 𝑥𝐵 ↔ ∀𝑥𝐶𝑗𝐴 𝑥𝐵))
5250feq2d 6722 . . . . . . 7 (𝑐 = 𝐶 → (𝑓:𝑐𝐴𝑓:𝐶𝐴))
5350raleqdv 3326 . . . . . . 7 (𝑐 = 𝐶 → (∀𝑥𝑐 𝑥𝐷 ↔ ∀𝑥𝐶 𝑥𝐷))
5452, 53anbi12d 632 . . . . . 6 (𝑐 = 𝐶 → ((𝑓:𝑐𝐴 ∧ ∀𝑥𝑐 𝑥𝐷) ↔ (𝑓:𝐶𝐴 ∧ ∀𝑥𝐶 𝑥𝐷)))
5554exbidv 1921 . . . . 5 (𝑐 = 𝐶 → (∃𝑓(𝑓:𝑐𝐴 ∧ ∀𝑥𝑐 𝑥𝐷) ↔ ∃𝑓(𝑓:𝐶𝐴 ∧ ∀𝑥𝐶 𝑥𝐷)))
5651, 55imbi12d 344 . . . 4 (𝑐 = 𝐶 → ((∀𝑥𝑐𝑗𝐴 𝑥𝐵 → ∃𝑓(𝑓:𝑐𝐴 ∧ ∀𝑥𝑐 𝑥𝐷)) ↔ (∀𝑥𝐶𝑗𝐴 𝑥𝐵 → ∃𝑓(𝑓:𝐶𝐴 ∧ ∀𝑥𝐶 𝑥𝐷))))
57 acunirnmpt2f.d . . . . . 6 𝑗𝐷
5857nfcri 2897 . . . . 5 𝑗 𝑥𝐷
59 vex 3484 . . . . 5 𝑐 ∈ V
60 acunirnmpt2f.3 . . . . . 6 (𝑗 = (𝑓𝑥) → 𝐵 = 𝐷)
6160eleq2d 2827 . . . . 5 (𝑗 = (𝑓𝑥) → (𝑥𝐵𝑥𝐷))
6238, 58, 59, 61ac6sf2 32634 . . . 4 (∀𝑥𝑐𝑗𝐴 𝑥𝐵 → ∃𝑓(𝑓:𝑐𝐴 ∧ ∀𝑥𝑐 𝑥𝐷))
6356, 62vtoclg 3554 . . 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 206  wa 395   = wceq 1540  wex 1779  wcel 2108  wnfc 2890  wne 2940  wral 3061  wrex 3070  Vcvv 3480  csb 3899  c0 4333   cuni 4907   ciun 4991  cmpt 5225  ran crn 5686  wf 6557  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-reg 9632  ax-inf2 9681  ax-ac2 10503
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-en 8986  df-r1 9804  df-rank 9805  df-card 9979  df-ac 10156
This theorem is referenced by:  aciunf1lem  32672
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