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Theorem elrnmpt1d 40183
Description: Elementhood in an image set. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
elrnmpt1d.1 𝐹 = (𝑥𝐴𝐵)
elrnmpt1d.2 (𝜑𝑥𝐴)
elrnmpt1d.3 (𝜑𝐵𝑉)
Assertion
Ref Expression
elrnmpt1d (𝜑𝐵 ∈ ran 𝐹)

Proof of Theorem elrnmpt1d
StepHypRef Expression
1 elrnmpt1d.2 . 2 (𝜑𝑥𝐴)
2 elrnmpt1d.3 . 2 (𝜑𝐵𝑉)
3 elrnmpt1d.1 . . 3 𝐹 = (𝑥𝐴𝐵)
43elrnmpt1 5578 . 2 ((𝑥𝐴𝐵𝑉) → 𝐵 ∈ ran 𝐹)
51, 2, 4syl2anc 580 1 (𝜑𝐵 ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wcel 2157  cmpt 4922  ran crn 5313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-mpt 4923  df-cnv 5320  df-dm 5322  df-rn 5323
This theorem is referenced by:  rnmptss2  40219  rnmptssbi  40224  supminfxrrnmpt  40444
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