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Theorem acunirnmpt 29324
 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 477 . . . . . 6 ((((𝜑𝑦𝐶) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
2 simplll 797 . . . . . . 7 ((((𝜑𝑦𝐶) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → 𝜑)
3 simplr 791 . . . . . . 7 ((((𝜑𝑦𝐶) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → 𝑗𝐴)
4 acunirnmpt.1 . . . . . . 7 ((𝜑𝑗𝐴) → 𝐵 ≠ ∅)
52, 3, 4syl2anc 692 . . . . . 6 ((((𝜑𝑦𝐶) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → 𝐵 ≠ ∅)
61, 5eqnetrd 2857 . . . . 5 ((((𝜑𝑦𝐶) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → 𝑦 ≠ ∅)
7 acunirnmpt.2 . . . . . . . . 9 𝐶 = ran (𝑗𝐴𝐵)
87eleq2i 2690 . . . . . . . 8 (𝑦𝐶𝑦 ∈ ran (𝑗𝐴𝐵))
9 vex 3192 . . . . . . . . 9 𝑦 ∈ V
10 eqid 2621 . . . . . . . . . 10 (𝑗𝐴𝐵) = (𝑗𝐴𝐵)
1110elrnmpt 5337 . . . . . . . . 9 (𝑦 ∈ V → (𝑦 ∈ ran (𝑗𝐴𝐵) ↔ ∃𝑗𝐴 𝑦 = 𝐵))
129, 11ax-mp 5 . . . . . . . 8 (𝑦 ∈ ran (𝑗𝐴𝐵) ↔ ∃𝑗𝐴 𝑦 = 𝐵)
138, 12bitri 264 . . . . . . 7 (𝑦𝐶 ↔ ∃𝑗𝐴 𝑦 = 𝐵)
1413biimpi 206 . . . . . 6 (𝑦𝐶 → ∃𝑗𝐴 𝑦 = 𝐵)
1514adantl 482 . . . . 5 ((𝜑𝑦𝐶) → ∃𝑗𝐴 𝑦 = 𝐵)
166, 15r19.29a 3072 . . . 4 ((𝜑𝑦𝐶) → 𝑦 ≠ ∅)
1716ralrimiva 2961 . . 3 (𝜑 → ∀𝑦𝐶 𝑦 ≠ ∅)
18 acunirnmpt.0 . . . . . 6 (𝜑𝐴𝑉)
19 mptexg 6444 . . . . . 6 (𝐴𝑉 → (𝑗𝐴𝐵) ∈ V)
20 rnexg 7052 . . . . . 6 ((𝑗𝐴𝐵) ∈ V → ran (𝑗𝐴𝐵) ∈ V)
2118, 19, 203syl 18 . . . . 5 (𝜑 → ran (𝑗𝐴𝐵) ∈ V)
227, 21syl5eqel 2702 . . . 4 (𝜑𝐶 ∈ V)
23 raleq 3130 . . . . . 6 (𝑐 = 𝐶 → (∀𝑦𝑐 𝑦 ≠ ∅ ↔ ∀𝑦𝐶 𝑦 ≠ ∅))
24 id 22 . . . . . . . . 9 (𝑐 = 𝐶𝑐 = 𝐶)
25 unieq 4415 . . . . . . . . 9 (𝑐 = 𝐶 𝑐 = 𝐶)
2624, 25feq23d 6002 . . . . . . . 8 (𝑐 = 𝐶 → (𝑓:𝑐 𝑐𝑓:𝐶 𝐶))
27 raleq 3130 . . . . . . . 8 (𝑐 = 𝐶 → (∀𝑦𝑐 (𝑓𝑦) ∈ 𝑦 ↔ ∀𝑦𝐶 (𝑓𝑦) ∈ 𝑦))
2826, 27anbi12d 746 . . . . . . 7 (𝑐 = 𝐶 → ((𝑓:𝑐 𝑐 ∧ ∀𝑦𝑐 (𝑓𝑦) ∈ 𝑦) ↔ (𝑓:𝐶 𝐶 ∧ ∀𝑦𝐶 (𝑓𝑦) ∈ 𝑦)))
2928exbidv 1847 . . . . . 6 (𝑐 = 𝐶 → (∃𝑓(𝑓:𝑐 𝑐 ∧ ∀𝑦𝑐 (𝑓𝑦) ∈ 𝑦) ↔ ∃𝑓(𝑓:𝐶 𝐶 ∧ ∀𝑦𝐶 (𝑓𝑦) ∈ 𝑦)))
3023, 29imbi12d 334 . . . . 5 (𝑐 = 𝐶 → ((∀𝑦𝑐 𝑦 ≠ ∅ → ∃𝑓(𝑓:𝑐 𝑐 ∧ ∀𝑦𝑐 (𝑓𝑦) ∈ 𝑦)) ↔ (∀𝑦𝐶 𝑦 ≠ ∅ → ∃𝑓(𝑓:𝐶 𝐶 ∧ ∀𝑦𝐶 (𝑓𝑦) ∈ 𝑦))))
31 vex 3192 . . . . . 6 𝑐 ∈ V
3231ac5b 9252 . . . . 5 (∀𝑦𝑐 𝑦 ≠ ∅ → ∃𝑓(𝑓:𝑐 𝑐 ∧ ∀𝑦𝑐 (𝑓𝑦) ∈ 𝑦))
3330, 32vtoclg 3255 . . . 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 481 . . . . . . 7 (((𝜑𝑦𝐶) ∧ (𝑓𝑦) ∈ 𝑦) → ∃𝑗𝐴 𝑦 = 𝐵)
37 simpllr 798 . . . . . . . . . 10 (((((𝜑𝑦𝐶) ∧ (𝑓𝑦) ∈ 𝑦) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → (𝑓𝑦) ∈ 𝑦)
38 simpr 477 . . . . . . . . . 10 (((((𝜑𝑦𝐶) ∧ (𝑓𝑦) ∈ 𝑦) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
3937, 38eleqtrd 2700 . . . . . . . . 9 (((((𝜑𝑦𝐶) ∧ (𝑓𝑦) ∈ 𝑦) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → (𝑓𝑦) ∈ 𝐵)
4039ex 450 . . . . . . . 8 ((((𝜑𝑦𝐶) ∧ (𝑓𝑦) ∈ 𝑦) ∧ 𝑗𝐴) → (𝑦 = 𝐵 → (𝑓𝑦) ∈ 𝐵))
4140reximdva 3012 . . . . . . 7 (((𝜑𝑦𝐶) ∧ (𝑓𝑦) ∈ 𝑦) → (∃𝑗𝐴 𝑦 = 𝐵 → ∃𝑗𝐴 (𝑓𝑦) ∈ 𝐵))
4236, 41mpd 15 . . . . . 6 (((𝜑𝑦𝐶) ∧ (𝑓𝑦) ∈ 𝑦) → ∃𝑗𝐴 (𝑓𝑦) ∈ 𝐵)
4342ex 450 . . . . 5 ((𝜑𝑦𝐶) → ((𝑓𝑦) ∈ 𝑦 → ∃𝑗𝐴 (𝑓𝑦) ∈ 𝐵))
4443ralimdva 2957 . . . 4 (𝜑 → (∀𝑦𝐶 (𝑓𝑦) ∈ 𝑦 → ∀𝑦𝐶𝑗𝐴 (𝑓𝑦) ∈ 𝐵))
4544anim2d 588 . . 3 (𝜑 → ((𝑓:𝐶 𝐶 ∧ ∀𝑦𝐶 (𝑓𝑦) ∈ 𝑦) → (𝑓:𝐶 𝐶 ∧ ∀𝑦𝐶𝑗𝐴 (𝑓𝑦) ∈ 𝐵)))
4645eximdv 1843 . 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 196   ∧ wa 384   = wceq 1480  ∃wex 1701   ∈ wcel 1987   ≠ wne 2790  ∀wral 2907  ∃wrex 2908  Vcvv 3189  ∅c0 3896  ∪ cuni 4407   ↦ cmpt 4678  ran crn 5080  ⟶wf 5848  ‘cfv 5852 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-ac2 9237 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-wrecs 7359  df-recs 7420  df-en 7908  df-card 8717  df-ac 8891 This theorem is referenced by: (None)
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