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Theorem fmptd2f 41868
 Description: Domain and codomain of the mapping operation; deduction form. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
fmptd2f.1 𝑥𝜑
fmptd2f.2 ((𝜑𝑥𝐴) → 𝐵𝐶)
Assertion
Ref Expression
fmptd2f (𝜑 → (𝑥𝐴𝐵):𝐴𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem fmptd2f
StepHypRef Expression
1 fmptd2f.1 . 2 𝑥𝜑
2 fmptd2f.2 . 2 ((𝜑𝑥𝐴) → 𝐵𝐶)
3 eqid 2801 . 2 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
41, 2, 3fmptdf 6862 1 (𝜑 → (𝑥𝐴𝐵):𝐴𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  Ⅎwnf 1785   ∈ wcel 2112   ↦ cmpt 5113  ⟶wf 6324 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336 This theorem is referenced by:  climinf2mpt  42353  climinfmpt  42354  limsupvaluzmpt  42356  limsupre2mpt  42369  limsupre3mpt  42373  limsupreuzmpt  42378  supcnvlimsupmpt  42380  liminfvalxrmpt  42425  liminflbuz2  42454  sge0z  43011  smflimsupmpt  43457  smfliminfmpt  43460
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