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Theorem acunirnmpt2 30320
 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 ≠ ∅)
acunirnmpt2.2 𝐶 = ran (𝑗𝐴𝐵)
acunirnmpt2.3 (𝑗 = (𝑓𝑥) → 𝐵 = 𝐷)
Assertion
Ref Expression
acunirnmpt2 (𝜑 → ∃𝑓(𝑓:𝐶𝐴 ∧ ∀𝑥𝐶 𝑥𝐷))
Distinct variable groups:   𝑓,𝑗,𝑥,𝐴   𝐵,𝑓   𝐶,𝑓,𝑗,𝑥   𝐷,𝑗   𝜑,𝑓,𝑗,𝑥
Allowed substitution hints:   𝐵(𝑥,𝑗)   𝐷(𝑥,𝑓)   𝑉(𝑥,𝑓,𝑗)

Proof of Theorem acunirnmpt2
Dummy variables 𝑐 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplr 765 . . . . . 6 ((((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦) → 𝑦 ∈ ran (𝑗𝐴𝐵))
2 vex 3503 . . . . . . 7 𝑦 ∈ V
3 eqid 2826 . . . . . . . 8 (𝑗𝐴𝐵) = (𝑗𝐴𝐵)
43elrnmpt 5827 . . . . . . 7 (𝑦 ∈ V → (𝑦 ∈ ran (𝑗𝐴𝐵) ↔ ∃𝑗𝐴 𝑦 = 𝐵))
52, 4ax-mp 5 . . . . . 6 (𝑦 ∈ ran (𝑗𝐴𝐵) ↔ ∃𝑗𝐴 𝑦 = 𝐵)
61, 5sylib 219 . . . . 5 ((((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦) → ∃𝑗𝐴 𝑦 = 𝐵)
7 nfv 1908 . . . . . . . 8 𝑗(𝜑𝑥𝐶)
8 nfcv 2982 . . . . . . . . 9 𝑗𝑦
9 nfmpt1 5161 . . . . . . . . . 10 𝑗(𝑗𝐴𝐵)
109nfrn 5823 . . . . . . . . 9 𝑗ran (𝑗𝐴𝐵)
118, 10nfel 2997 . . . . . . . 8 𝑗 𝑦 ∈ ran (𝑗𝐴𝐵)
127, 11nfan 1893 . . . . . . 7 𝑗((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵))
13 nfv 1908 . . . . . . 7 𝑗 𝑥𝑦
1412, 13nfan 1893 . . . . . 6 𝑗(((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦)
15 simpllr 772 . . . . . . . . 9 ((((((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → 𝑥𝑦)
16 simpr 485 . . . . . . . . 9 ((((((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
1715, 16eleqtrd 2920 . . . . . . . 8 ((((((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦) ∧ 𝑗𝐴) ∧ 𝑦 = 𝐵) → 𝑥𝐵)
1817ex 413 . . . . . . 7 (((((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦) ∧ 𝑗𝐴) → (𝑦 = 𝐵𝑥𝐵))
1918ex 413 . . . . . 6 ((((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦) → (𝑗𝐴 → (𝑦 = 𝐵𝑥𝐵)))
2014, 19reximdai 3316 . . . . 5 ((((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦) → (∃𝑗𝐴 𝑦 = 𝐵 → ∃𝑗𝐴 𝑥𝐵))
216, 20mpd 15 . . . 4 ((((𝜑𝑥𝐶) ∧ 𝑦 ∈ ran (𝑗𝐴𝐵)) ∧ 𝑥𝑦) → ∃𝑗𝐴 𝑥𝐵)
22 acunirnmpt2.2 . . . . . . . 8 𝐶 = ran (𝑗𝐴𝐵)
2322eleq2i 2909 . . . . . . 7 (𝑥𝐶𝑥 ran (𝑗𝐴𝐵))
2423biimpi 217 . . . . . 6 (𝑥𝐶𝑥 ran (𝑗𝐴𝐵))
25 eluni2 4841 . . . . . 6 (𝑥 ran (𝑗𝐴𝐵) ↔ ∃𝑦 ∈ ran (𝑗𝐴𝐵)𝑥𝑦)
2624, 25sylib 219 . . . . 5 (𝑥𝐶 → ∃𝑦 ∈ ran (𝑗𝐴𝐵)𝑥𝑦)
2726adantl 482 . . . 4 ((𝜑𝑥𝐶) → ∃𝑦 ∈ ran (𝑗𝐴𝐵)𝑥𝑦)
2821, 27r19.29a 3294 . . 3 ((𝜑𝑥𝐶) → ∃𝑗𝐴 𝑥𝐵)
2928ralrimiva 3187 . 2 (𝜑 → ∀𝑥𝐶𝑗𝐴 𝑥𝐵)
30 acunirnmpt.0 . . . . 5 (𝜑𝐴𝑉)
31 mptexg 6979 . . . . 5 (𝐴𝑉 → (𝑗𝐴𝐵) ∈ V)
32 rnexg 7602 . . . . 5 ((𝑗𝐴𝐵) ∈ V → ran (𝑗𝐴𝐵) ∈ V)
33 uniexg 7457 . . . . 5 (ran (𝑗𝐴𝐵) ∈ V → ran (𝑗𝐴𝐵) ∈ V)
3430, 31, 32, 334syl 19 . . . 4 (𝜑 ran (𝑗𝐴𝐵) ∈ V)
3522, 34eqeltrid 2922 . . 3 (𝜑𝐶 ∈ V)
36 id 22 . . . . . 6 (𝑐 = 𝐶𝑐 = 𝐶)
3736raleqdv 3421 . . . . 5 (𝑐 = 𝐶 → (∀𝑥𝑐𝑗𝐴 𝑥𝐵 ↔ ∀𝑥𝐶𝑗𝐴 𝑥𝐵))
3836feq2d 6497 . . . . . . 7 (𝑐 = 𝐶 → (𝑓:𝑐𝐴𝑓:𝐶𝐴))
3936raleqdv 3421 . . . . . . 7 (𝑐 = 𝐶 → (∀𝑥𝑐 𝑥𝐷 ↔ ∀𝑥𝐶 𝑥𝐷))
4038, 39anbi12d 630 . . . . . 6 (𝑐 = 𝐶 → ((𝑓:𝑐𝐴 ∧ ∀𝑥𝑐 𝑥𝐷) ↔ (𝑓:𝐶𝐴 ∧ ∀𝑥𝐶 𝑥𝐷)))
4140exbidv 1915 . . . . 5 (𝑐 = 𝐶 → (∃𝑓(𝑓:𝑐𝐴 ∧ ∀𝑥𝑐 𝑥𝐷) ↔ ∃𝑓(𝑓:𝐶𝐴 ∧ ∀𝑥𝐶 𝑥𝐷)))
4237, 41imbi12d 346 . . . 4 (𝑐 = 𝐶 → ((∀𝑥𝑐𝑗𝐴 𝑥𝐵 → ∃𝑓(𝑓:𝑐𝐴 ∧ ∀𝑥𝑐 𝑥𝐷)) ↔ (∀𝑥𝐶𝑗𝐴 𝑥𝐵 → ∃𝑓(𝑓:𝐶𝐴 ∧ ∀𝑥𝐶 𝑥𝐷))))
43 vex 3503 . . . . 5 𝑐 ∈ V
44 acunirnmpt2.3 . . . . . 6 (𝑗 = (𝑓𝑥) → 𝐵 = 𝐷)
4544eleq2d 2903 . . . . 5 (𝑗 = (𝑓𝑥) → (𝑥𝐵𝑥𝐷))
4643, 45ac6s 9895 . . . 4 (∀𝑥𝑐𝑗𝐴 𝑥𝐵 → ∃𝑓(𝑓:𝑐𝐴 ∧ ∀𝑥𝑐 𝑥𝐷))
4742, 46vtoclg 3573 . . 3 (𝐶 ∈ V → (∀𝑥𝐶𝑗𝐴 𝑥𝐵 → ∃𝑓(𝑓:𝐶𝐴 ∧ ∀𝑥𝐶 𝑥𝐷)))
4835, 47syl 17 . 2 (𝜑 → (∀𝑥𝐶𝑗𝐴 𝑥𝐵 → ∃𝑓(𝑓:𝐶𝐴 ∧ ∀𝑥𝐶 𝑥𝐷)))
4929, 48mpd 15 1 (𝜑 → ∃𝑓(𝑓:𝐶𝐴 ∧ ∀𝑥𝐶 𝑥𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1530  ∃wex 1773   ∈ wcel 2107   ≠ wne 3021  ∀wral 3143  ∃wrex 3144  Vcvv 3500  ∅c0 4295  ∪ cuni 4837   ↦ cmpt 5143  ran crn 5555  ⟶wf 6348  ‘cfv 6352 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-reg 9045  ax-inf2 9093  ax-ac2 9874 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-iin 4920  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-isom 6361  df-riota 7106  df-om 7569  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-en 8499  df-r1 9182  df-rank 9183  df-card 9357  df-ac 9531 This theorem is referenced by: (None)
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