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Theorem fvmptelrn 38423
 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 2610 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
32fmpt 6289 . . 3 (∀𝑥𝐴 𝐵𝐶 ↔ (𝑥𝐴𝐵):𝐴𝐶)
41, 3sylibr 223 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
54r19.21bi 2916 1 ((𝜑𝑥𝐴) → 𝐵𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977  ∀wral 2896   ↦ cmpt 4643  ⟶wf 5800 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812 This theorem is referenced by:  smfpimioompt  39671
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