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Theorem elrnmptdv 5915
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 3582 . 2 (𝜑 → ∃𝑥𝐴 𝐷 = 𝐵)
4 elrnmptdv.3 . . 3 (𝜑𝐷𝑉)
5 elrnmptdv.1 . . . 4 𝐹 = (𝑥𝐴𝐵)
65elrnmpt 5909 . . 3 (𝐷𝑉 → (𝐷 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐷 = 𝐵))
74, 6syl 17 . 2 (𝜑 → (𝐷 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐷 = 𝐵))
83, 7mpbird 256 1 (𝜑𝐷 ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wrex 3071  cmpt 5186  ran crn 5632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5166  df-mpt 5187  df-cnv 5639  df-dm 5641  df-rn 5642
This theorem is referenced by:  cycsubggend  18989  nsgqusf1olem3  32076  zart0  32329  rr-elrnmpt3d  42423
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