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Theorem acunirnmpt 30008
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 479 . . . . . 6 ((((𝜑𝑦𝐶) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
2 simplll 793 . . . . . . 7 ((((𝜑𝑦𝐶) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → 𝜑)
3 simplr 787 . . . . . . 7 ((((𝜑𝑦𝐶) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → 𝑗𝐴)
4 acunirnmpt.1 . . . . . . 7 ((𝜑𝑗𝐴) → 𝐵 ≠ ∅)
52, 3, 4syl2anc 581 . . . . . 6 ((((𝜑𝑦𝐶) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → 𝐵 ≠ ∅)
61, 5eqnetrd 3066 . . . . 5 ((((𝜑𝑦𝐶) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → 𝑦 ≠ ∅)
7 acunirnmpt.2 . . . . . . . . 9 𝐶 = ran (𝑗𝐴𝐵)
87eleq2i 2898 . . . . . . . 8 (𝑦𝐶𝑦 ∈ ran (𝑗𝐴𝐵))
9 vex 3417 . . . . . . . . 9 𝑦 ∈ V
10 eqid 2825 . . . . . . . . . 10 (𝑗𝐴𝐵) = (𝑗𝐴𝐵)
1110elrnmpt 5605 . . . . . . . . 9 (𝑦 ∈ V → (𝑦 ∈ ran (𝑗𝐴𝐵) ↔ ∃𝑗𝐴 𝑦 = 𝐵))
129, 11ax-mp 5 . . . . . . . 8 (𝑦 ∈ ran (𝑗𝐴𝐵) ↔ ∃𝑗𝐴 𝑦 = 𝐵)
138, 12bitri 267 . . . . . . 7 (𝑦𝐶 ↔ ∃𝑗𝐴 𝑦 = 𝐵)
1413biimpi 208 . . . . . 6 (𝑦𝐶 → ∃𝑗𝐴 𝑦 = 𝐵)
1514adantl 475 . . . . 5 ((𝜑𝑦𝐶) → ∃𝑗𝐴 𝑦 = 𝐵)
166, 15r19.29a 3288 . . . 4 ((𝜑𝑦𝐶) → 𝑦 ≠ ∅)
1716ralrimiva 3175 . . 3 (𝜑 → ∀𝑦𝐶 𝑦 ≠ ∅)
18 acunirnmpt.0 . . . . . 6 (𝜑𝐴𝑉)
19 mptexg 6740 . . . . . 6 (𝐴𝑉 → (𝑗𝐴𝐵) ∈ V)
20 rnexg 7359 . . . . . 6 ((𝑗𝐴𝐵) ∈ V → ran (𝑗𝐴𝐵) ∈ V)
2118, 19, 203syl 18 . . . . 5 (𝜑 → ran (𝑗𝐴𝐵) ∈ V)
227, 21syl5eqel 2910 . . . 4 (𝜑𝐶 ∈ V)
23 raleq 3350 . . . . . 6 (𝑐 = 𝐶 → (∀𝑦𝑐 𝑦 ≠ ∅ ↔ ∀𝑦𝐶 𝑦 ≠ ∅))
24 id 22 . . . . . . . . 9 (𝑐 = 𝐶𝑐 = 𝐶)
25 unieq 4666 . . . . . . . . 9 (𝑐 = 𝐶 𝑐 = 𝐶)
2624, 25feq23d 6273 . . . . . . . 8 (𝑐 = 𝐶 → (𝑓:𝑐 𝑐𝑓:𝐶 𝐶))
27 raleq 3350 . . . . . . . 8 (𝑐 = 𝐶 → (∀𝑦𝑐 (𝑓𝑦) ∈ 𝑦 ↔ ∀𝑦𝐶 (𝑓𝑦) ∈ 𝑦))
2826, 27anbi12d 626 . . . . . . 7 (𝑐 = 𝐶 → ((𝑓:𝑐 𝑐 ∧ ∀𝑦𝑐 (𝑓𝑦) ∈ 𝑦) ↔ (𝑓:𝐶 𝐶 ∧ ∀𝑦𝐶 (𝑓𝑦) ∈ 𝑦)))
2928exbidv 2022 . . . . . 6 (𝑐 = 𝐶 → (∃𝑓(𝑓:𝑐 𝑐 ∧ ∀𝑦𝑐 (𝑓𝑦) ∈ 𝑦) ↔ ∃𝑓(𝑓:𝐶 𝐶 ∧ ∀𝑦𝐶 (𝑓𝑦) ∈ 𝑦)))
3023, 29imbi12d 336 . . . . 5 (𝑐 = 𝐶 → ((∀𝑦𝑐 𝑦 ≠ ∅ → ∃𝑓(𝑓:𝑐 𝑐 ∧ ∀𝑦𝑐 (𝑓𝑦) ∈ 𝑦)) ↔ (∀𝑦𝐶 𝑦 ≠ ∅ → ∃𝑓(𝑓:𝐶 𝐶 ∧ ∀𝑦𝐶 (𝑓𝑦) ∈ 𝑦))))
31 vex 3417 . . . . . 6 𝑐 ∈ V
3231ac5b 9615 . . . . 5 (∀𝑦𝑐 𝑦 ≠ ∅ → ∃𝑓(𝑓:𝑐 𝑐 ∧ ∀𝑦𝑐 (𝑓𝑦) ∈ 𝑦))
3330, 32vtoclg 3482 . . . 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 474 . . . . . . 7 (((𝜑𝑦𝐶) ∧ (𝑓𝑦) ∈ 𝑦) → ∃𝑗𝐴 𝑦 = 𝐵)
37 simpllr 795 . . . . . . . . . 10 (((((𝜑𝑦𝐶) ∧ (𝑓𝑦) ∈ 𝑦) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → (𝑓𝑦) ∈ 𝑦)
38 simpr 479 . . . . . . . . . 10 (((((𝜑𝑦𝐶) ∧ (𝑓𝑦) ∈ 𝑦) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
3937, 38eleqtrd 2908 . . . . . . . . 9 (((((𝜑𝑦𝐶) ∧ (𝑓𝑦) ∈ 𝑦) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → (𝑓𝑦) ∈ 𝐵)
4039ex 403 . . . . . . . 8 ((((𝜑𝑦𝐶) ∧ (𝑓𝑦) ∈ 𝑦) ∧ 𝑗𝐴) → (𝑦 = 𝐵 → (𝑓𝑦) ∈ 𝐵))
4140reximdva 3225 . . . . . . 7 (((𝜑𝑦𝐶) ∧ (𝑓𝑦) ∈ 𝑦) → (∃𝑗𝐴 𝑦 = 𝐵 → ∃𝑗𝐴 (𝑓𝑦) ∈ 𝐵))
4236, 41mpd 15 . . . . . 6 (((𝜑𝑦𝐶) ∧ (𝑓𝑦) ∈ 𝑦) → ∃𝑗𝐴 (𝑓𝑦) ∈ 𝐵)
4342ex 403 . . . . 5 ((𝜑𝑦𝐶) → ((𝑓𝑦) ∈ 𝑦 → ∃𝑗𝐴 (𝑓𝑦) ∈ 𝐵))
4443ralimdva 3171 . . . 4 (𝜑 → (∀𝑦𝐶 (𝑓𝑦) ∈ 𝑦 → ∀𝑦𝐶𝑗𝐴 (𝑓𝑦) ∈ 𝐵))
4544anim2d 607 . . 3 (𝜑 → ((𝑓:𝐶 𝐶 ∧ ∀𝑦𝐶 (𝑓𝑦) ∈ 𝑦) → (𝑓:𝐶 𝐶 ∧ ∀𝑦𝐶𝑗𝐴 (𝑓𝑦) ∈ 𝐵)))
4645eximdv 2018 . 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 198  wa 386   = wceq 1658  wex 1880  wcel 2166  wne 2999  wral 3117  wrex 3118  Vcvv 3414  c0 4144   cuni 4658  cmpt 4952  ran crn 5343  wf 6119  cfv 6123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-ac2 9600
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-se 5302  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-isom 6132  df-riota 6866  df-wrecs 7672  df-recs 7734  df-en 8223  df-card 9078  df-ac 9252
This theorem is referenced by: (None)
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