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Theorem elrnmptdv 5979
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 5972 . . 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 1537  wcel 2106  wrex 3068  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:  cycsubggend  19236  nsgqusf1olem3  33423  zart0  33840  rr-elrnmpt3d  44198
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