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Theorem acunirnmpt 32669
Description: Axiom of choice for the union of the range of a mapping to function. (Contributed by Thierry Arnoux, 6-Nov-2019.)
Hypotheses
Ref Expression
acunirnmpt.0 (𝜑𝐴𝑉)
acunirnmpt.1 ((𝜑𝑗𝐴) → 𝐵 ≠ ∅)
acunirnmpt.2 𝐶 = ran (𝑗𝐴𝐵)
Assertion
Ref Expression
acunirnmpt (𝜑 → ∃𝑓(𝑓:𝐶 𝐶 ∧ ∀𝑦𝐶𝑗𝐴 (𝑓𝑦) ∈ 𝐵))
Distinct variable groups:   𝐴,𝑗   𝑓,𝑗,𝑦,𝐶   𝜑,𝑓,𝑗,𝑦
Allowed substitution hints:   𝐴(𝑦,𝑓)   𝐵(𝑦,𝑓,𝑗)   𝑉(𝑦,𝑓,𝑗)

Proof of Theorem acunirnmpt
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 simpr 484 . . . . . 6 ((((𝜑𝑦𝐶) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
2 simplll 775 . . . . . . 7 ((((𝜑𝑦𝐶) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → 𝜑)
3 simplr 769 . . . . . . 7 ((((𝜑𝑦𝐶) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → 𝑗𝐴)
4 acunirnmpt.1 . . . . . . 7 ((𝜑𝑗𝐴) → 𝐵 ≠ ∅)
52, 3, 4syl2anc 584 . . . . . 6 ((((𝜑𝑦𝐶) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → 𝐵 ≠ ∅)
61, 5eqnetrd 3008 . . . . 5 ((((𝜑𝑦𝐶) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → 𝑦 ≠ ∅)
7 acunirnmpt.2 . . . . . . . . 9 𝐶 = ran (𝑗𝐴𝐵)
87eleq2i 2833 . . . . . . . 8 (𝑦𝐶𝑦 ∈ ran (𝑗𝐴𝐵))
9 vex 3484 . . . . . . . . 9 𝑦 ∈ V
10 eqid 2737 . . . . . . . . . 10 (𝑗𝐴𝐵) = (𝑗𝐴𝐵)
1110elrnmpt 5969 . . . . . . . . 9 (𝑦 ∈ V → (𝑦 ∈ ran (𝑗𝐴𝐵) ↔ ∃𝑗𝐴 𝑦 = 𝐵))
129, 11ax-mp 5 . . . . . . . 8 (𝑦 ∈ ran (𝑗𝐴𝐵) ↔ ∃𝑗𝐴 𝑦 = 𝐵)
138, 12bitri 275 . . . . . . 7 (𝑦𝐶 ↔ ∃𝑗𝐴 𝑦 = 𝐵)
1413biimpi 216 . . . . . 6 (𝑦𝐶 → ∃𝑗𝐴 𝑦 = 𝐵)
1514adantl 481 . . . . 5 ((𝜑𝑦𝐶) → ∃𝑗𝐴 𝑦 = 𝐵)
166, 15r19.29a 3162 . . . 4 ((𝜑𝑦𝐶) → 𝑦 ≠ ∅)
1716ralrimiva 3146 . . 3 (𝜑 → ∀𝑦𝐶 𝑦 ≠ ∅)
18 acunirnmpt.0 . . . . . 6 (𝜑𝐴𝑉)
19 mptexg 7241 . . . . . 6 (𝐴𝑉 → (𝑗𝐴𝐵) ∈ V)
20 rnexg 7924 . . . . . 6 ((𝑗𝐴𝐵) ∈ V → ran (𝑗𝐴𝐵) ∈ V)
2118, 19, 203syl 18 . . . . 5 (𝜑 → ran (𝑗𝐴𝐵) ∈ V)
227, 21eqeltrid 2845 . . . 4 (𝜑𝐶 ∈ V)
23 raleq 3323 . . . . . 6 (𝑐 = 𝐶 → (∀𝑦𝑐 𝑦 ≠ ∅ ↔ ∀𝑦𝐶 𝑦 ≠ ∅))
24 id 22 . . . . . . . . 9 (𝑐 = 𝐶𝑐 = 𝐶)
25 unieq 4918 . . . . . . . . 9 (𝑐 = 𝐶 𝑐 = 𝐶)
2624, 25feq23d 6731 . . . . . . . 8 (𝑐 = 𝐶 → (𝑓:𝑐 𝑐𝑓:𝐶 𝐶))
27 raleq 3323 . . . . . . . 8 (𝑐 = 𝐶 → (∀𝑦𝑐 (𝑓𝑦) ∈ 𝑦 ↔ ∀𝑦𝐶 (𝑓𝑦) ∈ 𝑦))
2826, 27anbi12d 632 . . . . . . 7 (𝑐 = 𝐶 → ((𝑓:𝑐 𝑐 ∧ ∀𝑦𝑐 (𝑓𝑦) ∈ 𝑦) ↔ (𝑓:𝐶 𝐶 ∧ ∀𝑦𝐶 (𝑓𝑦) ∈ 𝑦)))
2928exbidv 1921 . . . . . 6 (𝑐 = 𝐶 → (∃𝑓(𝑓:𝑐 𝑐 ∧ ∀𝑦𝑐 (𝑓𝑦) ∈ 𝑦) ↔ ∃𝑓(𝑓:𝐶 𝐶 ∧ ∀𝑦𝐶 (𝑓𝑦) ∈ 𝑦)))
3023, 29imbi12d 344 . . . . 5 (𝑐 = 𝐶 → ((∀𝑦𝑐 𝑦 ≠ ∅ → ∃𝑓(𝑓:𝑐 𝑐 ∧ ∀𝑦𝑐 (𝑓𝑦) ∈ 𝑦)) ↔ (∀𝑦𝐶 𝑦 ≠ ∅ → ∃𝑓(𝑓:𝐶 𝐶 ∧ ∀𝑦𝐶 (𝑓𝑦) ∈ 𝑦))))
31 vex 3484 . . . . . 6 𝑐 ∈ V
3231ac5b 10518 . . . . 5 (∀𝑦𝑐 𝑦 ≠ ∅ → ∃𝑓(𝑓:𝑐 𝑐 ∧ ∀𝑦𝑐 (𝑓𝑦) ∈ 𝑦))
3330, 32vtoclg 3554 . . . 4 (𝐶 ∈ V → (∀𝑦𝐶 𝑦 ≠ ∅ → ∃𝑓(𝑓:𝐶 𝐶 ∧ ∀𝑦𝐶 (𝑓𝑦) ∈ 𝑦)))
3422, 33syl 17 . . 3 (𝜑 → (∀𝑦𝐶 𝑦 ≠ ∅ → ∃𝑓(𝑓:𝐶 𝐶 ∧ ∀𝑦𝐶 (𝑓𝑦) ∈ 𝑦)))
3517, 34mpd 15 . 2 (𝜑 → ∃𝑓(𝑓:𝐶 𝐶 ∧ ∀𝑦𝐶 (𝑓𝑦) ∈ 𝑦))
3615adantr 480 . . . . . . 7 (((𝜑𝑦𝐶) ∧ (𝑓𝑦) ∈ 𝑦) → ∃𝑗𝐴 𝑦 = 𝐵)
37 simpllr 776 . . . . . . . . . 10 (((((𝜑𝑦𝐶) ∧ (𝑓𝑦) ∈ 𝑦) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → (𝑓𝑦) ∈ 𝑦)
38 simpr 484 . . . . . . . . . 10 (((((𝜑𝑦𝐶) ∧ (𝑓𝑦) ∈ 𝑦) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
3937, 38eleqtrd 2843 . . . . . . . . 9 (((((𝜑𝑦𝐶) ∧ (𝑓𝑦) ∈ 𝑦) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → (𝑓𝑦) ∈ 𝐵)
4039ex 412 . . . . . . . 8 ((((𝜑𝑦𝐶) ∧ (𝑓𝑦) ∈ 𝑦) ∧ 𝑗𝐴) → (𝑦 = 𝐵 → (𝑓𝑦) ∈ 𝐵))
4140reximdva 3168 . . . . . . 7 (((𝜑𝑦𝐶) ∧ (𝑓𝑦) ∈ 𝑦) → (∃𝑗𝐴 𝑦 = 𝐵 → ∃𝑗𝐴 (𝑓𝑦) ∈ 𝐵))
4236, 41mpd 15 . . . . . 6 (((𝜑𝑦𝐶) ∧ (𝑓𝑦) ∈ 𝑦) → ∃𝑗𝐴 (𝑓𝑦) ∈ 𝐵)
4342ex 412 . . . . 5 ((𝜑𝑦𝐶) → ((𝑓𝑦) ∈ 𝑦 → ∃𝑗𝐴 (𝑓𝑦) ∈ 𝐵))
4443ralimdva 3167 . . . 4 (𝜑 → (∀𝑦𝐶 (𝑓𝑦) ∈ 𝑦 → ∀𝑦𝐶𝑗𝐴 (𝑓𝑦) ∈ 𝐵))
4544anim2d 612 . . 3 (𝜑 → ((𝑓:𝐶 𝐶 ∧ ∀𝑦𝐶 (𝑓𝑦) ∈ 𝑦) → (𝑓:𝐶 𝐶 ∧ ∀𝑦𝐶𝑗𝐴 (𝑓𝑦) ∈ 𝐵)))
4645eximdv 1917 . 2 (𝜑 → (∃𝑓(𝑓:𝐶 𝐶 ∧ ∀𝑦𝐶 (𝑓𝑦) ∈ 𝑦) → ∃𝑓(𝑓:𝐶 𝐶 ∧ ∀𝑦𝐶𝑗𝐴 (𝑓𝑦) ∈ 𝐵)))
4735, 46mpd 15 1 (𝜑 → ∃𝑓(𝑓:𝐶 𝐶 ∧ ∀𝑦𝐶𝑗𝐴 (𝑓𝑦) ∈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108  wne 2940  wral 3061  wrex 3070  Vcvv 3480  c0 4333   cuni 4907  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-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-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-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-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-en 8986  df-card 9979  df-ac 10156
This theorem is referenced by: (None)
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