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Theorem acunirnmpt 29800
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 480 . . . . . 6 ((((𝜑𝑦𝐶) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
2 simplll 812 . . . . . . 7 ((((𝜑𝑦𝐶) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → 𝜑)
3 simplr 806 . . . . . . 7 ((((𝜑𝑦𝐶) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → 𝑗𝐴)
4 acunirnmpt.1 . . . . . . 7 ((𝜑𝑗𝐴) → 𝐵 ≠ ∅)
52, 3, 4syl2anc 693 . . . . . 6 ((((𝜑𝑦𝐶) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → 𝐵 ≠ ∅)
61, 5eqnetrd 3008 . . . . 5 ((((𝜑𝑦𝐶) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → 𝑦 ≠ ∅)
7 acunirnmpt.2 . . . . . . . . 9 𝐶 = ran (𝑗𝐴𝐵)
87eleq2i 2840 . . . . . . . 8 (𝑦𝐶𝑦 ∈ ran (𝑗𝐴𝐵))
9 vex 3351 . . . . . . . . 9 𝑦 ∈ V
10 eqid 2769 . . . . . . . . . 10 (𝑗𝐴𝐵) = (𝑗𝐴𝐵)
1110elrnmpt 5509 . . . . . . . . 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 474 . . . . 5 ((𝜑𝑦𝐶) → ∃𝑗𝐴 𝑦 = 𝐵)
166, 15r19.29a 3224 . . . 4 ((𝜑𝑦𝐶) → 𝑦 ≠ ∅)
1716ralrimiva 3113 . . 3 (𝜑 → ∀𝑦𝐶 𝑦 ≠ ∅)
18 acunirnmpt.0 . . . . . 6 (𝜑𝐴𝑉)
19 mptexg 6626 . . . . . 6 (𝐴𝑉 → (𝑗𝐴𝐵) ∈ V)
20 rnexg 7243 . . . . . 6 ((𝑗𝐴𝐵) ∈ V → ran (𝑗𝐴𝐵) ∈ V)
2118, 19, 203syl 18 . . . . 5 (𝜑 → ran (𝑗𝐴𝐵) ∈ V)
227, 21syl5eqel 2852 . . . 4 (𝜑𝐶 ∈ V)
23 raleq 3285 . . . . . 6 (𝑐 = 𝐶 → (∀𝑦𝑐 𝑦 ≠ ∅ ↔ ∀𝑦𝐶 𝑦 ≠ ∅))
24 id 22 . . . . . . . . 9 (𝑐 = 𝐶𝑐 = 𝐶)
25 unieq 4579 . . . . . . . . 9 (𝑐 = 𝐶 𝑐 = 𝐶)
2624, 25feq23d 6179 . . . . . . . 8 (𝑐 = 𝐶 → (𝑓:𝑐 𝑐𝑓:𝐶 𝐶))
27 raleq 3285 . . . . . . . 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 2000 . . . . . 6 (𝑐 = 𝐶 → (∃𝑓(𝑓:𝑐 𝑐 ∧ ∀𝑦𝑐 (𝑓𝑦) ∈ 𝑦) ↔ ∃𝑓(𝑓:𝐶 𝐶 ∧ ∀𝑦𝐶 (𝑓𝑦) ∈ 𝑦)))
3023, 29imbi12d 333 . . . . 5 (𝑐 = 𝐶 → ((∀𝑦𝑐 𝑦 ≠ ∅ → ∃𝑓(𝑓:𝑐 𝑐 ∧ ∀𝑦𝑐 (𝑓𝑦) ∈ 𝑦)) ↔ (∀𝑦𝐶 𝑦 ≠ ∅ → ∃𝑓(𝑓:𝐶 𝐶 ∧ ∀𝑦𝐶 (𝑓𝑦) ∈ 𝑦))))
31 vex 3351 . . . . . 6 𝑐 ∈ V
3231ac5b 9500 . . . . 5 (∀𝑦𝑐 𝑦 ≠ ∅ → ∃𝑓(𝑓:𝑐 𝑐 ∧ ∀𝑦𝑐 (𝑓𝑦) ∈ 𝑦))
3330, 32vtoclg 3414 . . . 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 473 . . . . . . 7 (((𝜑𝑦𝐶) ∧ (𝑓𝑦) ∈ 𝑦) → ∃𝑗𝐴 𝑦 = 𝐵)
37 simpllr 814 . . . . . . . . . 10 (((((𝜑𝑦𝐶) ∧ (𝑓𝑦) ∈ 𝑦) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → (𝑓𝑦) ∈ 𝑦)
38 simpr 480 . . . . . . . . . 10 (((((𝜑𝑦𝐶) ∧ (𝑓𝑦) ∈ 𝑦) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
3937, 38eleqtrd 2850 . . . . . . . . 9 (((((𝜑𝑦𝐶) ∧ (𝑓𝑦) ∈ 𝑦) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → (𝑓𝑦) ∈ 𝐵)
4039ex 448 . . . . . . . 8 ((((𝜑𝑦𝐶) ∧ (𝑓𝑦) ∈ 𝑦) ∧ 𝑗𝐴) → (𝑦 = 𝐵 → (𝑓𝑦) ∈ 𝐵))
4140reximdva 3163 . . . . . . 7 (((𝜑𝑦𝐶) ∧ (𝑓𝑦) ∈ 𝑦) → (∃𝑗𝐴 𝑦 = 𝐵 → ∃𝑗𝐴 (𝑓𝑦) ∈ 𝐵))
4236, 41mpd 15 . . . . . 6 (((𝜑𝑦𝐶) ∧ (𝑓𝑦) ∈ 𝑦) → ∃𝑗𝐴 (𝑓𝑦) ∈ 𝐵)
4342ex 448 . . . . 5 ((𝜑𝑦𝐶) → ((𝑓𝑦) ∈ 𝑦 → ∃𝑗𝐴 (𝑓𝑦) ∈ 𝐵))
4443ralimdva 3109 . . . 4 (𝜑 → (∀𝑦𝐶 (𝑓𝑦) ∈ 𝑦 → ∀𝑦𝐶𝑗𝐴 (𝑓𝑦) ∈ 𝐵))
4544anim2d 591 . . 3 (𝜑 → ((𝑓:𝐶 𝐶 ∧ ∀𝑦𝐶 (𝑓𝑦) ∈ 𝑦) → (𝑓:𝐶 𝐶 ∧ ∀𝑦𝐶𝑗𝐴 (𝑓𝑦) ∈ 𝐵)))
4645eximdv 1996 . 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 383   = wceq 1629  wex 1850  wcel 2143  wne 2941  wral 3059  wrex 3060  Vcvv 3348  c0 4060   cuni 4571  cmpt 4860  ran crn 5249  wf 6026  cfv 6030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1868  ax-4 1883  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2145  ax-9 2152  ax-10 2172  ax-11 2188  ax-12 2201  ax-13 2406  ax-ext 2749  ax-rep 4901  ax-sep 4911  ax-nul 4919  ax-pow 4970  ax-pr 5033  ax-un 7094  ax-ac2 9485
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1070  df-3an 1071  df-tru 1632  df-ex 1851  df-nf 1856  df-sb 2048  df-eu 2620  df-mo 2621  df-clab 2756  df-cleq 2762  df-clel 2765  df-nfc 2900  df-ne 2942  df-ral 3064  df-rex 3065  df-reu 3066  df-rmo 3067  df-rab 3068  df-v 3350  df-sbc 3585  df-csb 3680  df-dif 3723  df-un 3725  df-in 3727  df-ss 3734  df-pss 3736  df-nul 4061  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4572  df-int 4609  df-iun 4653  df-br 4784  df-opab 4844  df-mpt 4861  df-tr 4884  df-id 5156  df-eprel 5161  df-po 5169  df-so 5170  df-fr 5207  df-se 5208  df-we 5209  df-xp 5254  df-rel 5255  df-cnv 5256  df-co 5257  df-dm 5258  df-rn 5259  df-res 5260  df-ima 5261  df-pred 5822  df-ord 5868  df-on 5869  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-isom 6039  df-riota 6752  df-wrecs 7557  df-recs 7619  df-en 8108  df-card 8963  df-ac 9137
This theorem is referenced by: (None)
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