Users' Mathboxes Mathbox for Rohan Ridenour < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rr-elrnmpt3d Structured version   Visualization version   GIF version

Theorem rr-elrnmpt3d 44198
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 2741 . 2 ((𝜑𝑥 = 𝐶) → 𝐷 = 𝐵)
61, 2, 3, 5elrnmptdv 5979 1 (𝜑𝐷 ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  cmpt 5231  ran crn 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-mpt 5232  df-cnv 5697  df-dm 5699  df-rn 5700
This theorem is referenced by:  mnurndlem1  44277
  Copyright terms: Public domain W3C validator