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Theorem rr-elrnmpt3d 43703
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 2731 . 2 ((𝜑𝑥 = 𝐶) → 𝐷 = 𝐵)
61, 2, 3, 5elrnmptdv 5959 1 (𝜑𝐷 ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  cmpt 5226  ran crn 5673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-mpt 5227  df-cnv 5680  df-dm 5682  df-rn 5683
This theorem is referenced by:  mnurndlem1  43783
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