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Theorem elrnmptdv 4805
 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 2802 . 2 (𝜑 → ∃𝑥𝐴 𝐷 = 𝐵)
4 elrnmptdv.3 . . 3 (𝜑𝐷𝑉)
5 elrnmptdv.1 . . . 4 𝐹 = (𝑥𝐴𝐵)
65elrnmpt 4800 . . 3 (𝐷𝑉 → (𝐷 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐷 = 𝐵))
74, 6syl 14 . 2 (𝜑 → (𝐷 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐷 = 𝐵))
83, 7mpbird 166 1 (𝜑𝐷 ∈ ran 𝐹)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332   ∈ wcel 1481  ∃wrex 2419   ↦ cmpt 3999  ran crn 4552 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2123  ax-sep 4056  ax-pow 4108  ax-pr 4142 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1738  df-eu 2004  df-mo 2005  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-rex 2424  df-v 2693  df-un 3082  df-in 3084  df-ss 3091  df-pw 3519  df-sn 3540  df-pr 3541  df-op 3543  df-br 3940  df-opab 4000  df-mpt 4001  df-cnv 4559  df-dm 4561  df-rn 4562 This theorem is referenced by: (None)
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