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Theorem acnlem 8815
 Description: Construct a mapping satisfying the consequent of isacn 8811. (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 6174 . . . . . 6 (𝑓𝑥) ⊆ ran 𝑓
2 simpr 477 . . . . . 6 ((𝑥𝐴𝐵 ∈ (𝑓𝑥)) → 𝐵 ∈ (𝑓𝑥))
31, 2sseldi 3581 . . . . 5 ((𝑥𝐴𝐵 ∈ (𝑓𝑥)) → 𝐵 ran 𝑓)
43ralimiaa 2946 . . . 4 (∀𝑥𝐴 𝐵 ∈ (𝑓𝑥) → ∀𝑥𝐴 𝐵 ran 𝑓)
5 eqid 2621 . . . . 5 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
65fmpt 6337 . . . 4 (∀𝑥𝐴 𝐵 ran 𝑓 ↔ (𝑥𝐴𝐵):𝐴 ran 𝑓)
74, 6sylib 208 . . 3 (∀𝑥𝐴 𝐵 ∈ (𝑓𝑥) → (𝑥𝐴𝐵):𝐴 ran 𝑓)
8 id 22 . . 3 (𝐴𝑉𝐴𝑉)
9 vex 3189 . . . . . 6 𝑓 ∈ V
109rnex 7047 . . . . 5 ran 𝑓 ∈ V
1110uniex 6906 . . . 4 ran 𝑓 ∈ V
12 fex2 7068 . . . 4 (((𝑥𝐴𝐵):𝐴 ran 𝑓𝐴𝑉 ran 𝑓 ∈ V) → (𝑥𝐴𝐵) ∈ V)
1311, 12mp3an3 1410 . . 3 (((𝑥𝐴𝐵):𝐴 ran 𝑓𝐴𝑉) → (𝑥𝐴𝐵) ∈ V)
147, 8, 13syl2anr 495 . 2 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐵 ∈ (𝑓𝑥)) → (𝑥𝐴𝐵) ∈ V)
155fvmpt2 6248 . . . . 5 ((𝑥𝐴𝐵 ∈ (𝑓𝑥)) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
1615, 2eqeltrd 2698 . . . 4 ((𝑥𝐴𝐵 ∈ (𝑓𝑥)) → ((𝑥𝐴𝐵)‘𝑥) ∈ (𝑓𝑥))
1716ralimiaa 2946 . . 3 (∀𝑥𝐴 𝐵 ∈ (𝑓𝑥) → ∀𝑥𝐴 ((𝑥𝐴𝐵)‘𝑥) ∈ (𝑓𝑥))
1817adantl 482 . 2 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐵 ∈ (𝑓𝑥)) → ∀𝑥𝐴 ((𝑥𝐴𝐵)‘𝑥) ∈ (𝑓𝑥))
19 nfmpt1 4707 . . . . 5 𝑥(𝑥𝐴𝐵)
2019nfeq2 2776 . . . 4 𝑥 𝑔 = (𝑥𝐴𝐵)
21 fveq1 6147 . . . . 5 (𝑔 = (𝑥𝐴𝐵) → (𝑔𝑥) = ((𝑥𝐴𝐵)‘𝑥))
2221eleq1d 2683 . . . 4 (𝑔 = (𝑥𝐴𝐵) → ((𝑔𝑥) ∈ (𝑓𝑥) ↔ ((𝑥𝐴𝐵)‘𝑥) ∈ (𝑓𝑥)))
2320, 22ralbid 2977 . . 3 (𝑔 = (𝑥𝐴𝐵) → (∀𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥) ↔ ∀𝑥𝐴 ((𝑥𝐴𝐵)‘𝑥) ∈ (𝑓𝑥)))
2423spcegv 3280 . 2 ((𝑥𝐴𝐵) ∈ V → (∀𝑥𝐴 ((𝑥𝐴𝐵)‘𝑥) ∈ (𝑓𝑥) → ∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥)))
2514, 18, 24sylc 65 1 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐵 ∈ (𝑓𝑥)) → ∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480  ∃wex 1701   ∈ wcel 1987  ∀wral 2907  Vcvv 3186  ∪ cuni 4402   ↦ cmpt 4673  ran crn 5075  ⟶wf 5843  ‘cfv 5847 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855 This theorem is referenced by:  numacn  8816  acndom  8818  acndom2  8821
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