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Theorem fvmptelrn 6987
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
fvmptelrn.1 (𝜑 → (𝑥𝐴𝐵):𝐴𝐶)
Assertion
Ref Expression
fvmptelrn ((𝜑𝑥𝐴) → 𝐵𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem fvmptelrn
StepHypRef Expression
1 fvmptelrn.1 . . 3 (𝜑 → (𝑥𝐴𝐵):𝐴𝐶)
2 eqid 2738 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
32fmpt 6984 . . 3 (∀𝑥𝐴 𝐵𝐶 ↔ (𝑥𝐴𝐵):𝐴𝐶)
41, 3sylibr 233 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
54r19.21bi 3134 1 ((𝜑𝑥𝐴) → 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  wral 3064  cmpt 5157  wf 6429
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-fun 6435  df-fn 6436  df-f 6437
This theorem is referenced by:  rlimmptrcl  15317  lo1mptrcl  15331  o1mptrcl  15332  frlmgsum  20979  uvcresum  21000  psrass1lemOLD  21143  psrass1lem  21146  txcnp  22771  ptcnp  22773  ptcn  22778  cnmpt11  22814  cnmpt1t  22816  cnmpt12  22818  cnmptkp  22831  cnmptk1  22832  cnmptkk  22834  cnmptk1p  22836  cnmptk2  22837  cnmpt1plusg  23238  cnmpt1vsca  23345  cnmpt1ds  24005  cncfcompt2  24071  cncfmpt2ss  24079  cnmpt1ip  24411  divcncf  24611  mbfmptcl  24800  i1fposd  24872  itgss3  24979  dvmptcl  25123  dvmptco  25136  dvle  25171  dvfsumle  25185  dvfsumge  25186  dvmptrecl  25188  itgparts  25211  itgsubstlem  25212  itgsubst  25213  ulmss  25556  ulmdvlem2  25560  itgulm2  25568  logtayl  25815  intlewftc  40069  cncfcompt  43424  cncficcgt0  43429  itgsubsticclem  43516  sge0iunmptlemre  43953  hoicvrrex  44094  smfadd  44300  smfpimioompt  44320  smfinfmpt  44352
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