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Theorem rr-elrnmpt3d 44170
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 2746 . 2 ((𝜑𝑥 = 𝐶) → 𝐷 = 𝐵)
61, 2, 3, 5elrnmptdv 5988 1 (𝜑𝐷 ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2108  cmpt 5249  ran crn 5701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-mpt 5250  df-cnv 5708  df-dm 5710  df-rn 5711
This theorem is referenced by:  mnurndlem1  44250
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