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Theorem acnlem 9071
Description: Construct a mapping satisfying the consequent of isacn 9067. (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 6358 . . . . . 6 (𝑓𝑥) ⊆ ran 𝑓
2 simpr 471 . . . . . 6 ((𝑥𝐴𝐵 ∈ (𝑓𝑥)) → 𝐵 ∈ (𝑓𝑥))
31, 2sseldi 3750 . . . . 5 ((𝑥𝐴𝐵 ∈ (𝑓𝑥)) → 𝐵 ran 𝑓)
43ralimiaa 3100 . . . 4 (∀𝑥𝐴 𝐵 ∈ (𝑓𝑥) → ∀𝑥𝐴 𝐵 ran 𝑓)
5 eqid 2771 . . . . 5 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
65fmpt 6523 . . . 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 3354 . . . . . 6 𝑓 ∈ V
109rnex 7247 . . . . 5 ran 𝑓 ∈ V
1110uniex 7100 . . . 4 ran 𝑓 ∈ V
12 fex2 7268 . . . 4 (((𝑥𝐴𝐵):𝐴 ran 𝑓𝐴𝑉 ran 𝑓 ∈ V) → (𝑥𝐴𝐵) ∈ V)
1311, 12mp3an3 1561 . . 3 (((𝑥𝐴𝐵):𝐴 ran 𝑓𝐴𝑉) → (𝑥𝐴𝐵) ∈ V)
147, 8, 13syl2anr 584 . 2 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐵 ∈ (𝑓𝑥)) → (𝑥𝐴𝐵) ∈ V)
155fvmpt2 6433 . . . . 5 ((𝑥𝐴𝐵 ∈ (𝑓𝑥)) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
1615, 2eqeltrd 2850 . . . 4 ((𝑥𝐴𝐵 ∈ (𝑓𝑥)) → ((𝑥𝐴𝐵)‘𝑥) ∈ (𝑓𝑥))
1716ralimiaa 3100 . . 3 (∀𝑥𝐴 𝐵 ∈ (𝑓𝑥) → ∀𝑥𝐴 ((𝑥𝐴𝐵)‘𝑥) ∈ (𝑓𝑥))
1817adantl 467 . 2 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐵 ∈ (𝑓𝑥)) → ∀𝑥𝐴 ((𝑥𝐴𝐵)‘𝑥) ∈ (𝑓𝑥))
19 nfmpt1 4881 . . . . 5 𝑥(𝑥𝐴𝐵)
2019nfeq2 2929 . . . 4 𝑥 𝑔 = (𝑥𝐴𝐵)
21 fveq1 6331 . . . . 5 (𝑔 = (𝑥𝐴𝐵) → (𝑔𝑥) = ((𝑥𝐴𝐵)‘𝑥))
2221eleq1d 2835 . . . 4 (𝑔 = (𝑥𝐴𝐵) → ((𝑔𝑥) ∈ (𝑓𝑥) ↔ ((𝑥𝐴𝐵)‘𝑥) ∈ (𝑓𝑥)))
2320, 22ralbid 3132 . . 3 (𝑔 = (𝑥𝐴𝐵) → (∀𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥) ↔ ∀𝑥𝐴 ((𝑥𝐴𝐵)‘𝑥) ∈ (𝑓𝑥)))
2423spcegv 3445 . 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 382   = wceq 1631  wex 1852  wcel 2145  wral 3061  Vcvv 3351   cuni 4574  cmpt 4863  ran crn 5250  wf 6027  cfv 6031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039
This theorem is referenced by:  numacn  9072  acndom  9074  acndom2  9077
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