Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  acunirnmpt2f Structured version   Visualization version   GIF version

Theorem acunirnmpt2f 29589
 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 807 . . . . . 6 ((((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦) → 𝑦 ∈ ran (𝑗𝐴𝐵))
2 vex 3234 . . . . . . 7 𝑦 ∈ V
3 eqid 2651 . . . . . . . 8 (𝑗𝐴𝐵) = (𝑗𝐴𝐵)
43elrnmpt 5404 . . . . . . 7 (𝑦 ∈ V → (𝑦 ∈ ran (𝑗𝐴𝐵) ↔ ∃𝑗𝐴 𝑦 = 𝐵))
52, 4ax-mp 5 . . . . . 6 (𝑦 ∈ ran (𝑗𝐴𝐵) ↔ ∃𝑗𝐴 𝑦 = 𝐵)
61, 5sylib 208 . . . . 5 ((((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦) → ∃𝑗𝐴 𝑦 = 𝐵)
7 nfv 1883 . . . . . . . . 9 𝑗𝜑
8 acunirnmpt2f.c . . . . . . . . . 10 𝑗𝐶
98nfcri 2787 . . . . . . . . 9 𝑗 𝑥𝐶
107, 9nfan 1868 . . . . . . . 8 𝑗(𝜑𝑥𝐶)
11 nfcv 2793 . . . . . . . . 9 𝑗𝑦
12 nfmpt1 4780 . . . . . . . . . 10 𝑗(𝑗𝐴𝐵)
1312nfrn 5400 . . . . . . . . 9 𝑗ran (𝑗𝐴𝐵)
1411, 13nfel 2806 . . . . . . . 8 𝑗 𝑦 ∈ ran (𝑗𝐴𝐵)
1510, 14nfan 1868 . . . . . . 7 𝑗((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵))
16 nfv 1883 . . . . . . 7 𝑗 𝑥𝑦
1715, 16nfan 1868 . . . . . 6 𝑗(((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦)
18 simpllr 815 . . . . . . . . 9 ((((((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → 𝑥𝑦)
19 simpr 476 . . . . . . . . 9 ((((((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
2018, 19eleqtrd 2732 . . . . . . . 8 ((((((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → 𝑥𝐵)
2120ex 449 . . . . . . 7 (((((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦) ∧ 𝑗𝐴) → (𝑦 = 𝐵𝑥𝐵))
2221ex 449 . . . . . 6 ((((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦) → (𝑗𝐴 → (𝑦 = 𝐵𝑥𝐵)))
2317, 22reximdai 3041 . . . . 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 2995 . . . . . . . . 9 (𝜑 → ∀𝑗𝐴 𝐵𝑊)
28 dfiun3g 5410 . . . . . . . . 9 (∀𝑗𝐴 𝐵𝑊 𝑗𝐴 𝐵 = ran (𝑗𝐴𝐵))
2927, 28syl 17 . . . . . . . 8 (𝜑 𝑗𝐴 𝐵 = ran (𝑗𝐴𝐵))
3025, 29syl5eq 2697 . . . . . . 7 (𝜑𝐶 = ran (𝑗𝐴𝐵))
3130eleq2d 2716 . . . . . 6 (𝜑 → (𝑥𝐶𝑥 ran (𝑗𝐴𝐵)))
3231biimpa 500 . . . . 5 ((𝜑𝑥𝐶) → 𝑥 ran (𝑗𝐴𝐵))
33 eluni2 4472 . . . . 5 (𝑥 ran (𝑗𝐴𝐵) ↔ ∃𝑦 ∈ ran (𝑗𝐴𝐵)𝑥𝑦)
3432, 33sylib 208 . . . 4 ((𝜑𝑥𝐶) → ∃𝑦 ∈ ran (𝑗𝐴𝐵)𝑥𝑦)
3524, 34r19.29a 3107 . . 3 ((𝜑𝑥𝐶) → ∃𝑗𝐴 𝑥𝐵)
3635ralrimiva 2995 . 2 (𝜑 → ∀𝑥𝐶𝑗𝐴 𝑥𝐵)
37 acunirnmpt.0 . . . . 5 (𝜑𝐴𝑉)
38 aciunf1lem.a . . . . . . 7 𝑗𝐴
39 nfcv 2793 . . . . . . 7 𝑘𝐴
40 nfcv 2793 . . . . . . 7 𝑘𝐵
41 nfcsb1v 3582 . . . . . . 7 𝑗𝑘 / 𝑗𝐵
42 csbeq1a 3575 . . . . . . 7 (𝑗 = 𝑘𝐵 = 𝑘 / 𝑗𝐵)
4338, 39, 40, 41, 42cbvmptf 4781 . . . . . 6 (𝑗𝐴𝐵) = (𝑘𝐴𝑘 / 𝑗𝐵)
44 mptexg 6525 . . . . . 6 (𝐴𝑉 → (𝑘𝐴𝑘 / 𝑗𝐵) ∈ V)
4543, 44syl5eqel 2734 . . . . 5 (𝐴𝑉 → (𝑗𝐴𝐵) ∈ V)
46 rnexg 7140 . . . . 5 ((𝑗𝐴𝐵) ∈ V → ran (𝑗𝐴𝐵) ∈ V)
47 uniexg 6997 . . . . 5 (ran (𝑗𝐴𝐵) ∈ V → ran (𝑗𝐴𝐵) ∈ V)
4837, 45, 46, 474syl 19 . . . 4 (𝜑 ran (𝑗𝐴𝐵) ∈ V)
4930, 48eqeltrd 2730 . . 3 (𝜑𝐶 ∈ V)
50 id 22 . . . . . 6 (𝑐 = 𝐶𝑐 = 𝐶)
5150raleqdv 3174 . . . . 5 (𝑐 = 𝐶 → (∀𝑥𝑐𝑗𝐴 𝑥𝐵 ↔ ∀𝑥𝐶𝑗𝐴 𝑥𝐵))
5250feq2d 6069 . . . . . . 7 (𝑐 = 𝐶 → (𝑓:𝑐𝐴𝑓:𝐶𝐴))
5350raleqdv 3174 . . . . . . 7 (𝑐 = 𝐶 → (∀𝑥𝑐 𝑥𝐷 ↔ ∀𝑥𝐶 𝑥𝐷))
5452, 53anbi12d 747 . . . . . 6 (𝑐 = 𝐶 → ((𝑓:𝑐𝐴 ∧ ∀𝑥𝑐 𝑥𝐷) ↔ (𝑓:𝐶𝐴 ∧ ∀𝑥𝐶 𝑥𝐷)))
5554exbidv 1890 . . . . 5 (𝑐 = 𝐶 → (∃𝑓(𝑓:𝑐𝐴 ∧ ∀𝑥𝑐 𝑥𝐷) ↔ ∃𝑓(𝑓:𝐶𝐴 ∧ ∀𝑥𝐶 𝑥𝐷)))
5651, 55imbi12d 333 . . . 4 (𝑐 = 𝐶 → ((∀𝑥𝑐𝑗𝐴 𝑥𝐵 → ∃𝑓(𝑓:𝑐𝐴 ∧ ∀𝑥𝑐 𝑥𝐷)) ↔ (∀𝑥𝐶𝑗𝐴 𝑥𝐵 → ∃𝑓(𝑓:𝐶𝐴 ∧ ∀𝑥𝐶 𝑥𝐷))))
57 acunirnmpt2f.d . . . . . 6 𝑗𝐷
5857nfcri 2787 . . . . 5 𝑗 𝑥𝐷
59 vex 3234 . . . . 5 𝑐 ∈ V
60 acunirnmpt2f.3 . . . . . 6 (𝑗 = (𝑓𝑥) → 𝐵 = 𝐷)
6160eleq2d 2716 . . . . 5 (𝑗 = (𝑓𝑥) → (𝑥𝐵𝑥𝐷))
6238, 58, 59, 61ac6sf2 29557 . . . 4 (∀𝑥𝑐𝑗𝐴 𝑥𝐵 → ∃𝑓(𝑓:𝑐𝐴 ∧ ∀𝑥𝑐 𝑥𝐷))
6356, 62vtoclg 3297 . . 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 196   ∧ wa 383   = wceq 1523  ∃wex 1744   ∈ wcel 2030  Ⅎwnfc 2780   ≠ wne 2823  ∀wral 2941  ∃wrex 2942  Vcvv 3231  ⦋csb 3566  ∅c0 3948  ∪ cuni 4468  ∪ ciun 4552   ↦ cmpt 4762  ran crn 5144  ⟶wf 5922  ‘cfv 5926 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-reg 8538  ax-inf2 8576  ax-ac2 9323 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-en 7998  df-r1 8665  df-rank 8666  df-card 8803  df-ac 8977 This theorem is referenced by:  aciunf1lem  29590
 Copyright terms: Public domain W3C validator