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Theorem fvmptelrn 6871
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 2821 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
32fmpt 6868 . . 3 (∀𝑥𝐴 𝐵𝐶 ↔ (𝑥𝐴𝐵):𝐴𝐶)
41, 3sylibr 236 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
54r19.21bi 3208 1 ((𝜑𝑥𝐴) → 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wcel 2110  wral 3138  cmpt 5138  wf 6345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-fv 6357
This theorem is referenced by:  rlimmptrcl  14958  lo1mptrcl  14972  o1mptrcl  14973  psrass1lem  20151  frlmgsum  20910  uvcresum  20931  txcnp  22222  ptcnp  22224  ptcn  22229  cnmpt11  22265  cnmpt1t  22267  cnmpt12  22269  cnmptkp  22282  cnmptk1  22283  cnmptkk  22285  cnmptk1p  22287  cnmptk2  22288  cnmpt1plusg  22689  cnmpt1vsca  22796  cnmpt1ds  23444  cncfmpt2ss  23517  cnmpt1ip  23844  divcncf  24042  mbfmptcl  24231  i1fposd  24302  itgss3  24409  dvmptcl  24550  dvmptco  24563  dvle  24598  dvfsumle  24612  dvfsumge  24613  dvmptrecl  24615  itgparts  24638  itgsubstlem  24639  itgsubst  24640  ulmss  24979  ulmdvlem2  24983  itgulm2  24991  logtayl  25237  cncfcompt  42159  cncficcgt0  42164  cncfcompt2  42175  itgsubsticclem  42253  sge0iunmptlemre  42691  hoicvrrex  42832  smfadd  43035  smfpimioompt  43055  smfsupmpt  43083  smfinfmpt  43087
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