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Theorem acnlem 10088
Description: Construct a mapping satisfying the consequent of isacn 10084. (Contributed by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
acnlem ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐵 ∈ (𝑓𝑥)) → ∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥))
Distinct variable groups:   𝑓,𝑔,𝑥,𝐴   𝐵,𝑔
Allowed substitution hints:   𝐵(𝑥,𝑓)   𝑉(𝑥,𝑓,𝑔)

Proof of Theorem acnlem
StepHypRef Expression
1 fvssunirn 6939 . . . . . 6 (𝑓𝑥) ⊆ ran 𝑓
2 simpr 484 . . . . . 6 ((𝑥𝐴𝐵 ∈ (𝑓𝑥)) → 𝐵 ∈ (𝑓𝑥))
31, 2sselid 3981 . . . . 5 ((𝑥𝐴𝐵 ∈ (𝑓𝑥)) → 𝐵 ran 𝑓)
43ralimiaa 3082 . . . 4 (∀𝑥𝐴 𝐵 ∈ (𝑓𝑥) → ∀𝑥𝐴 𝐵 ran 𝑓)
5 eqid 2737 . . . . 5 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
65fmpt 7130 . . . 4 (∀𝑥𝐴 𝐵 ran 𝑓 ↔ (𝑥𝐴𝐵):𝐴 ran 𝑓)
74, 6sylib 218 . . 3 (∀𝑥𝐴 𝐵 ∈ (𝑓𝑥) → (𝑥𝐴𝐵):𝐴 ran 𝑓)
8 id 22 . . 3 (𝐴𝑉𝐴𝑉)
9 vex 3484 . . . . . 6 𝑓 ∈ V
109rnex 7932 . . . . 5 ran 𝑓 ∈ V
1110uniex 7761 . . . 4 ran 𝑓 ∈ V
12 fex2 7958 . . . 4 (((𝑥𝐴𝐵):𝐴 ran 𝑓𝐴𝑉 ran 𝑓 ∈ V) → (𝑥𝐴𝐵) ∈ V)
1311, 12mp3an3 1452 . . 3 (((𝑥𝐴𝐵):𝐴 ran 𝑓𝐴𝑉) → (𝑥𝐴𝐵) ∈ V)
147, 8, 13syl2anr 597 . 2 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐵 ∈ (𝑓𝑥)) → (𝑥𝐴𝐵) ∈ V)
155fvmpt2 7027 . . . . 5 ((𝑥𝐴𝐵 ∈ (𝑓𝑥)) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
1615, 2eqeltrd 2841 . . . 4 ((𝑥𝐴𝐵 ∈ (𝑓𝑥)) → ((𝑥𝐴𝐵)‘𝑥) ∈ (𝑓𝑥))
1716ralimiaa 3082 . . 3 (∀𝑥𝐴 𝐵 ∈ (𝑓𝑥) → ∀𝑥𝐴 ((𝑥𝐴𝐵)‘𝑥) ∈ (𝑓𝑥))
1817adantl 481 . 2 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐵 ∈ (𝑓𝑥)) → ∀𝑥𝐴 ((𝑥𝐴𝐵)‘𝑥) ∈ (𝑓𝑥))
19 nfmpt1 5250 . . . 4 𝑥(𝑥𝐴𝐵)
2019nfeq2 2923 . . 3 𝑥 𝑔 = (𝑥𝐴𝐵)
21 fveq1 6905 . . . 4 (𝑔 = (𝑥𝐴𝐵) → (𝑔𝑥) = ((𝑥𝐴𝐵)‘𝑥))
2221eleq1d 2826 . . 3 (𝑔 = (𝑥𝐴𝐵) → ((𝑔𝑥) ∈ (𝑓𝑥) ↔ ((𝑥𝐴𝐵)‘𝑥) ∈ (𝑓𝑥)))
2320, 22ralbid 3273 . 2 (𝑔 = (𝑥𝐴𝐵) → (∀𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥) ↔ ∀𝑥𝐴 ((𝑥𝐴𝐵)‘𝑥) ∈ (𝑓𝑥)))
2414, 18, 23spcedv 3598 1 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐵 ∈ (𝑓𝑥)) → ∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wex 1779  wcel 2108  wral 3061  Vcvv 3480   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-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  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-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  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-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569
This theorem is referenced by:  numacn  10089  acndom  10091  acndom2  10094
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