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Theorem elrnmpt1d 41728
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 5817 . 2 ((𝑥𝐴𝐵𝑉) → 𝐵 ∈ ran 𝐹)
51, 2, 4syl2anc 587 1 (𝜑𝐵 ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  cmpt 5132  ran crn 5543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rex 3139  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-sn 4550  df-pr 4552  df-op 4556  df-br 5053  df-opab 5115  df-mpt 5133  df-cnv 5550  df-dm 5552  df-rn 5553
This theorem is referenced by:  rnmptbd2lem  41748  rnmptbdlem  41755  rnmptss2  41757  rnmptssbi  41762  supminfxrrnmpt  41973
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