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Theorem acnlem 9204
 Description: Construct a mapping satisfying the consequent of isacn 9200. (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 6475 . . . . . 6 (𝑓𝑥) ⊆ ran 𝑓
2 simpr 479 . . . . . 6 ((𝑥𝐴𝐵 ∈ (𝑓𝑥)) → 𝐵 ∈ (𝑓𝑥))
31, 2sseldi 3818 . . . . 5 ((𝑥𝐴𝐵 ∈ (𝑓𝑥)) → 𝐵 ran 𝑓)
43ralimiaa 3132 . . . 4 (∀𝑥𝐴 𝐵 ∈ (𝑓𝑥) → ∀𝑥𝐴 𝐵 ran 𝑓)
5 eqid 2777 . . . . 5 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
65fmpt 6644 . . . 4 (∀𝑥𝐴 𝐵 ran 𝑓 ↔ (𝑥𝐴𝐵):𝐴 ran 𝑓)
74, 6sylib 210 . . 3 (∀𝑥𝐴 𝐵 ∈ (𝑓𝑥) → (𝑥𝐴𝐵):𝐴 ran 𝑓)
8 id 22 . . 3 (𝐴𝑉𝐴𝑉)
9 vex 3400 . . . . . 6 𝑓 ∈ V
109rnex 7379 . . . . 5 ran 𝑓 ∈ V
1110uniex 7230 . . . 4 ran 𝑓 ∈ V
12 fex2 7400 . . . 4 (((𝑥𝐴𝐵):𝐴 ran 𝑓𝐴𝑉 ran 𝑓 ∈ V) → (𝑥𝐴𝐵) ∈ V)
1311, 12mp3an3 1523 . . 3 (((𝑥𝐴𝐵):𝐴 ran 𝑓𝐴𝑉) → (𝑥𝐴𝐵) ∈ V)
147, 8, 13syl2anr 590 . 2 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐵 ∈ (𝑓𝑥)) → (𝑥𝐴𝐵) ∈ V)
155fvmpt2 6552 . . . . 5 ((𝑥𝐴𝐵 ∈ (𝑓𝑥)) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
1615, 2eqeltrd 2858 . . . 4 ((𝑥𝐴𝐵 ∈ (𝑓𝑥)) → ((𝑥𝐴𝐵)‘𝑥) ∈ (𝑓𝑥))
1716ralimiaa 3132 . . 3 (∀𝑥𝐴 𝐵 ∈ (𝑓𝑥) → ∀𝑥𝐴 ((𝑥𝐴𝐵)‘𝑥) ∈ (𝑓𝑥))
1817adantl 475 . 2 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐵 ∈ (𝑓𝑥)) → ∀𝑥𝐴 ((𝑥𝐴𝐵)‘𝑥) ∈ (𝑓𝑥))
19 nfmpt1 4982 . . . 4 𝑥(𝑥𝐴𝐵)
2019nfeq2 2948 . . 3 𝑥 𝑔 = (𝑥𝐴𝐵)
21 fveq1 6445 . . . 4 (𝑔 = (𝑥𝐴𝐵) → (𝑔𝑥) = ((𝑥𝐴𝐵)‘𝑥))
2221eleq1d 2843 . . 3 (𝑔 = (𝑥𝐴𝐵) → ((𝑔𝑥) ∈ (𝑓𝑥) ↔ ((𝑥𝐴𝐵)‘𝑥) ∈ (𝑓𝑥)))
2320, 22ralbid 3164 . 2 (𝑔 = (𝑥𝐴𝐵) → (∀𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥) ↔ ∀𝑥𝐴 ((𝑥𝐴𝐵)‘𝑥) ∈ (𝑓𝑥)))
2414, 18, 23elabd 3559 1 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐵 ∈ (𝑓𝑥)) → ∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601  ∃wex 1823   ∈ wcel 2106  ∀wral 3089  Vcvv 3397  ∪ cuni 4671   ↦ cmpt 4965  ran crn 5356  ⟶wf 6131  ‘cfv 6135 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-fv 6143 This theorem is referenced by:  numacn  9205  acndom  9207  acndom2  9210
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