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Theorem acunirnmpt2f 32153
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 765 . . . . . 6 ((((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦) → 𝑦 ∈ ran (𝑗𝐴𝐵))
2 vex 3476 . . . . . . 7 𝑦 ∈ V
3 eqid 2730 . . . . . . . 8 (𝑗𝐴𝐵) = (𝑗𝐴𝐵)
43elrnmpt 5954 . . . . . . 7 (𝑦 ∈ V → (𝑦 ∈ ran (𝑗𝐴𝐵) ↔ ∃𝑗𝐴 𝑦 = 𝐵))
52, 4ax-mp 5 . . . . . 6 (𝑦 ∈ ran (𝑗𝐴𝐵) ↔ ∃𝑗𝐴 𝑦 = 𝐵)
61, 5sylib 217 . . . . 5 ((((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦) → ∃𝑗𝐴 𝑦 = 𝐵)
7 nfv 1915 . . . . . . . . 9 𝑗𝜑
8 acunirnmpt2f.c . . . . . . . . . 10 𝑗𝐶
98nfcri 2888 . . . . . . . . 9 𝑗 𝑥𝐶
107, 9nfan 1900 . . . . . . . 8 𝑗(𝜑𝑥𝐶)
11 nfcv 2901 . . . . . . . . 9 𝑗𝑦
12 nfmpt1 5255 . . . . . . . . . 10 𝑗(𝑗𝐴𝐵)
1312nfrn 5950 . . . . . . . . 9 𝑗ran (𝑗𝐴𝐵)
1411, 13nfel 2915 . . . . . . . 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 772 . . . . . . . . 9 ((((((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → 𝑥𝑦)
19 simpr 483 . . . . . . . . 9 ((((((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
2018, 19eleqtrd 2833 . . . . . . . 8 ((((((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → 𝑥𝐵)
2120ex 411 . . . . . . 7 (((((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦) ∧ 𝑗𝐴) → (𝑦 = 𝐵𝑥𝐵))
2221ex 411 . . . . . 6 ((((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦) → (𝑗𝐴 → (𝑦 = 𝐵𝑥𝐵)))
2317, 22reximdai 3256 . . . . 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 3144 . . . . . . . . 9 (𝜑 → ∀𝑗𝐴 𝐵𝑊)
28 dfiun3g 5962 . . . . . . . . 9 (∀𝑗𝐴 𝐵𝑊 𝑗𝐴 𝐵 = ran (𝑗𝐴𝐵))
2927, 28syl 17 . . . . . . . 8 (𝜑 𝑗𝐴 𝐵 = ran (𝑗𝐴𝐵))
3025, 29eqtrid 2782 . . . . . . 7 (𝜑𝐶 = ran (𝑗𝐴𝐵))
3130eleq2d 2817 . . . . . 6 (𝜑 → (𝑥𝐶𝑥 ran (𝑗𝐴𝐵)))
3231biimpa 475 . . . . 5 ((𝜑𝑥𝐶) → 𝑥 ran (𝑗𝐴𝐵))
33 eluni2 4911 . . . . 5 (𝑥 ran (𝑗𝐴𝐵) ↔ ∃𝑦 ∈ ran (𝑗𝐴𝐵)𝑥𝑦)
3432, 33sylib 217 . . . 4 ((𝜑𝑥𝐶) → ∃𝑦 ∈ ran (𝑗𝐴𝐵)𝑥𝑦)
3524, 34r19.29a 3160 . . 3 ((𝜑𝑥𝐶) → ∃𝑗𝐴 𝑥𝐵)
3635ralrimiva 3144 . 2 (𝜑 → ∀𝑥𝐶𝑗𝐴 𝑥𝐵)
37 acunirnmpt.0 . . . . 5 (𝜑𝐴𝑉)
38 aciunf1lem.a . . . . . . 7 𝑗𝐴
39 nfcv 2901 . . . . . . 7 𝑘𝐴
40 nfcv 2901 . . . . . . 7 𝑘𝐵
41 nfcsb1v 3917 . . . . . . 7 𝑗𝑘 / 𝑗𝐵
42 csbeq1a 3906 . . . . . . 7 (𝑗 = 𝑘𝐵 = 𝑘 / 𝑗𝐵)
4338, 39, 40, 41, 42cbvmptf 5256 . . . . . 6 (𝑗𝐴𝐵) = (𝑘𝐴𝑘 / 𝑗𝐵)
44 mptexg 7224 . . . . . 6 (𝐴𝑉 → (𝑘𝐴𝑘 / 𝑗𝐵) ∈ V)
4543, 44eqeltrid 2835 . . . . 5 (𝐴𝑉 → (𝑗𝐴𝐵) ∈ V)
46 rnexg 7897 . . . . 5 ((𝑗𝐴𝐵) ∈ V → ran (𝑗𝐴𝐵) ∈ V)
47 uniexg 7732 . . . . 5 (ran (𝑗𝐴𝐵) ∈ V → ran (𝑗𝐴𝐵) ∈ V)
4837, 45, 46, 474syl 19 . . . 4 (𝜑 ran (𝑗𝐴𝐵) ∈ V)
4930, 48eqeltrd 2831 . . 3 (𝜑𝐶 ∈ V)
50 id 22 . . . . . 6 (𝑐 = 𝐶𝑐 = 𝐶)
5150raleqdv 3323 . . . . 5 (𝑐 = 𝐶 → (∀𝑥𝑐𝑗𝐴 𝑥𝐵 ↔ ∀𝑥𝐶𝑗𝐴 𝑥𝐵))
5250feq2d 6702 . . . . . . 7 (𝑐 = 𝐶 → (𝑓:𝑐𝐴𝑓:𝐶𝐴))
5350raleqdv 3323 . . . . . . 7 (𝑐 = 𝐶 → (∀𝑥𝑐 𝑥𝐷 ↔ ∀𝑥𝐶 𝑥𝐷))
5452, 53anbi12d 629 . . . . . 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 343 . . . 4 (𝑐 = 𝐶 → ((∀𝑥𝑐𝑗𝐴 𝑥𝐵 → ∃𝑓(𝑓:𝑐𝐴 ∧ ∀𝑥𝑐 𝑥𝐷)) ↔ (∀𝑥𝐶𝑗𝐴 𝑥𝐵 → ∃𝑓(𝑓:𝐶𝐴 ∧ ∀𝑥𝐶 𝑥𝐷))))
57 acunirnmpt2f.d . . . . . 6 𝑗𝐷
5857nfcri 2888 . . . . 5 𝑗 𝑥𝐷
59 vex 3476 . . . . 5 𝑐 ∈ V
60 acunirnmpt2f.3 . . . . . 6 (𝑗 = (𝑓𝑥) → 𝐵 = 𝐷)
6160eleq2d 2817 . . . . 5 (𝑗 = (𝑓𝑥) → (𝑥𝐵𝑥𝐷))
6238, 58, 59, 61ac6sf2 32116 . . . 4 (∀𝑥𝑐𝑗𝐴 𝑥𝐵 → ∃𝑓(𝑓:𝑐𝐴 ∧ ∀𝑥𝑐 𝑥𝐷))
6356, 62vtoclg 3541 . . 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 205  wa 394   = wceq 1539  wex 1779  wcel 2104  wnfc 2881  wne 2938  wral 3059  wrex 3068  Vcvv 3472  csb 3892  c0 4321   cuni 4907   ciun 4996  cmpt 5230  ran crn 5676  wf 6538  cfv 6542
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-reg 9589  ax-inf2 9638  ax-ac2 10460
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7367  df-ov 7414  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-en 8942  df-r1 9761  df-rank 9762  df-card 9936  df-ac 10113
This theorem is referenced by:  aciunf1lem  32154
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