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Theorem rr-elrnmpt3d 40833
Description: Elementhood in an image set. (Contributed by Rohan Ridenour, 11-Aug-2023.)
Hypotheses
Ref Expression
rr-elrnmpt3d.1 𝐹 = (𝑥𝐴𝐵)
rr-elrnmpt3d.2 (𝜑𝐶𝐴)
rr-elrnmpt3d.3 (𝜑𝐷𝑉)
rr-elrnmpt3d.4 ((𝜑𝑥 = 𝐶) → 𝐵 = 𝐷)
Assertion
Ref Expression
rr-elrnmpt3d (𝜑𝐷 ∈ ran 𝐹)
Distinct variable groups:   𝑥,𝐷   𝑥,𝐴   𝑥,𝐶   𝜑,𝑥
Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem rr-elrnmpt3d
StepHypRef Expression
1 rr-elrnmpt3d.1 . 2 𝐹 = (𝑥𝐴𝐵)
2 rr-elrnmpt3d.2 . 2 (𝜑𝐶𝐴)
3 rr-elrnmpt3d.3 . 2 (𝜑𝐷𝑉)
4 rr-elrnmpt3d.4 . . 3 ((𝜑𝑥 = 𝐶) → 𝐵 = 𝐷)
54eqcomd 2830 . 2 ((𝜑𝑥 = 𝐶) → 𝐷 = 𝐵)
61, 2, 3, 5elrnmptdv 5820 1 (𝜑𝐷 ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = 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-ral 3138  df-rex 3139  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-mpt 5133  df-cnv 5550  df-dm 5552  df-rn 5553
This theorem is referenced by:  mnurndlem1  40909
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