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Theorem fvmptelrn 38937
Description: A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
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 2621 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
32fmpt 6347 . . 3 (∀𝑥𝐴 𝐵𝐶 ↔ (𝑥𝐴𝐵):𝐴𝐶)
41, 3sylibr 224 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
54r19.21bi 2928 1 ((𝜑𝑥𝐴) → 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  wral 2908  cmpt 4683  wf 5853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fv 5865
This theorem is referenced by:  smfpimioompt  40330  smfsupmpt  40358  smfinfmpt  40362
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