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Theorem elrnmptdv 5976
Description: Elementhood in the range of a function in maps-to notation, deduction form. (Contributed by Rohan Ridenour, 3-Aug-2023.)
Hypotheses
Ref Expression
elrnmptdv.1 𝐹 = (𝑥𝐴𝐵)
elrnmptdv.2 (𝜑𝐶𝐴)
elrnmptdv.3 (𝜑𝐷𝑉)
elrnmptdv.4 ((𝜑𝑥 = 𝐶) → 𝐷 = 𝐵)
Assertion
Ref Expression
elrnmptdv (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 elrnmptdv
StepHypRef Expression
1 elrnmptdv.4 . . 3 ((𝜑𝑥 = 𝐶) → 𝐷 = 𝐵)
2 elrnmptdv.2 . . 3 (𝜑𝐶𝐴)
31, 2rspcime 3627 . 2 (𝜑 → ∃𝑥𝐴 𝐷 = 𝐵)
4 elrnmptdv.3 . . 3 (𝜑𝐷𝑉)
5 elrnmptdv.1 . . . 4 𝐹 = (𝑥𝐴𝐵)
65elrnmpt 5969 . . 3 (𝐷𝑉 → (𝐷 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐷 = 𝐵))
74, 6syl 17 . 2 (𝜑 → (𝐷 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐷 = 𝐵))
83, 7mpbird 257 1 (𝜑𝐷 ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wrex 3070  cmpt 5225  ran crn 5686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-mpt 5226  df-cnv 5693  df-dm 5695  df-rn 5696
This theorem is referenced by:  cycsubggend  19223  nsgqusf1olem3  33443  zart0  33878  rr-elrnmpt3d  44221
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