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Theorem acnlem 9560
Description: Construct a mapping satisfying the consequent of isacn 9556. (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 6715 . . . . . 6 (𝑓𝑥) ⊆ ran 𝑓
2 simpr 488 . . . . . 6 ((𝑥𝐴𝐵 ∈ (𝑓𝑥)) → 𝐵 ∈ (𝑓𝑥))
31, 2sseldi 3885 . . . . 5 ((𝑥𝐴𝐵 ∈ (𝑓𝑥)) → 𝐵 ran 𝑓)
43ralimiaa 3075 . . . 4 (∀𝑥𝐴 𝐵 ∈ (𝑓𝑥) → ∀𝑥𝐴 𝐵 ran 𝑓)
5 eqid 2739 . . . . 5 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
65fmpt 6896 . . . 4 (∀𝑥𝐴 𝐵 ran 𝑓 ↔ (𝑥𝐴𝐵):𝐴 ran 𝑓)
74, 6sylib 221 . . 3 (∀𝑥𝐴 𝐵 ∈ (𝑓𝑥) → (𝑥𝐴𝐵):𝐴 ran 𝑓)
8 id 22 . . 3 (𝐴𝑉𝐴𝑉)
9 vex 3404 . . . . . 6 𝑓 ∈ V
109rnex 7655 . . . . 5 ran 𝑓 ∈ V
1110uniex 7497 . . . 4 ran 𝑓 ∈ V
12 fex2 7676 . . . 4 (((𝑥𝐴𝐵):𝐴 ran 𝑓𝐴𝑉 ran 𝑓 ∈ V) → (𝑥𝐴𝐵) ∈ V)
1311, 12mp3an3 1451 . . 3 (((𝑥𝐴𝐵):𝐴 ran 𝑓𝐴𝑉) → (𝑥𝐴𝐵) ∈ V)
147, 8, 13syl2anr 600 . 2 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐵 ∈ (𝑓𝑥)) → (𝑥𝐴𝐵) ∈ V)
155fvmpt2 6798 . . . . 5 ((𝑥𝐴𝐵 ∈ (𝑓𝑥)) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
1615, 2eqeltrd 2834 . . . 4 ((𝑥𝐴𝐵 ∈ (𝑓𝑥)) → ((𝑥𝐴𝐵)‘𝑥) ∈ (𝑓𝑥))
1716ralimiaa 3075 . . 3 (∀𝑥𝐴 𝐵 ∈ (𝑓𝑥) → ∀𝑥𝐴 ((𝑥𝐴𝐵)‘𝑥) ∈ (𝑓𝑥))
1817adantl 485 . 2 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐵 ∈ (𝑓𝑥)) → ∀𝑥𝐴 ((𝑥𝐴𝐵)‘𝑥) ∈ (𝑓𝑥))
19 nfmpt1 5138 . . . 4 𝑥(𝑥𝐴𝐵)
2019nfeq2 2917 . . 3 𝑥 𝑔 = (𝑥𝐴𝐵)
21 fveq1 6685 . . . 4 (𝑔 = (𝑥𝐴𝐵) → (𝑔𝑥) = ((𝑥𝐴𝐵)‘𝑥))
2221eleq1d 2818 . . 3 (𝑔 = (𝑥𝐴𝐵) → ((𝑔𝑥) ∈ (𝑓𝑥) ↔ ((𝑥𝐴𝐵)‘𝑥) ∈ (𝑓𝑥)))
2320, 22ralbid 3146 . 2 (𝑔 = (𝑥𝐴𝐵) → (∀𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥) ↔ ∀𝑥𝐴 ((𝑥𝐴𝐵)‘𝑥) ∈ (𝑓𝑥)))
2414, 18, 23spcedv 3505 1 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐵 ∈ (𝑓𝑥)) → ∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wex 1786  wcel 2114  wral 3054  Vcvv 3400   cuni 4806  cmpt 5120  ran crn 5536  wf 6345  cfv 6349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7491
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-iota 6307  df-fun 6351  df-fn 6352  df-f 6353  df-fv 6357
This theorem is referenced by:  numacn  9561  acndom  9563  acndom2  9566
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