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Theorem fvmptelcdm 7113
Description: The value of a function at a point of its domain belongs to its codomain. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypothesis
Ref Expression
fvmptelcdm.1 (𝜑 → (𝑥𝐴𝐵):𝐴𝐶)
Assertion
Ref Expression
fvmptelcdm ((𝜑𝑥𝐴) → 𝐵𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem fvmptelcdm
StepHypRef Expression
1 fvmptelcdm.1 . . 3 (𝜑 → (𝑥𝐴𝐵):𝐴𝐶)
2 eqid 2733 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
32fmpt 7110 . . 3 (∀𝑥𝐴 𝐵𝐶 ↔ (𝑥𝐴𝐵):𝐴𝐶)
41, 3sylibr 233 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
54r19.21bi 3249 1 ((𝜑𝑥𝐴) → 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  wral 3062  cmpt 5232  wf 6540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-fun 6546  df-fn 6547  df-f 6548
This theorem is referenced by:  rlimmptrcl  15552  lo1mptrcl  15566  o1mptrcl  15567  frlmgsum  21327  uvcresum  21348  psrass1lemOLD  21493  psrass1lem  21496  txcnp  23124  ptcnp  23126  ptcn  23131  cnmpt11  23167  cnmpt1t  23169  cnmpt12  23171  cnmptkp  23184  cnmptk1  23185  cnmptkk  23187  cnmptk1p  23189  cnmptk2  23190  cnmpt1plusg  23591  cnmpt1vsca  23698  cnmpt1ds  24358  cncfcompt2  24424  cncfmpt2ss  24432  cnmpt1ip  24764  divcncf  24964  mbfmptcl  25153  i1fposd  25225  itgss3  25332  dvmptcl  25476  dvmptco  25489  dvle  25524  dvfsumle  25538  dvfsumge  25539  dvmptrecl  25541  itgparts  25564  itgsubstlem  25565  itgsubst  25566  ulmss  25909  ulmdvlem2  25913  itgulm2  25921  logtayl  26168  gg-dvfsumle  35182  intlewftc  40926  cncfcompt  44599  cncficcgt0  44604  itgsubsticclem  44691  sge0iunmptlemre  45131  hoicvrrex  45272  smfadd  45481  smfpimioompt  45502  smfinfmpt  45535
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