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Theorem elrnmpt2d 4794
 Description: Elementhood in the range of a function in maps-to notation, deduction form. (Contributed by Rohan Ridenour, 3-Aug-2023.)
Hypotheses
Ref Expression
elrnmpt2d.1 𝐹 = (𝑥𝐴𝐵)
elrnmpt2d.2 (𝜑𝐶 ∈ ran 𝐹)
Assertion
Ref Expression
elrnmpt2d (𝜑 → ∃𝑥𝐴 𝐶 = 𝐵)
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem elrnmpt2d
StepHypRef Expression
1 elrnmpt2d.2 . 2 (𝜑𝐶 ∈ ran 𝐹)
2 elrnmpt2d.1 . . . 4 𝐹 = (𝑥𝐴𝐵)
32elrnmpt 4788 . . 3 (𝐶 ∈ ran 𝐹 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
43ibi 175 . 2 (𝐶 ∈ ran 𝐹 → ∃𝑥𝐴 𝐶 = 𝐵)
51, 4syl 14 1 (𝜑 → ∃𝑥𝐴 𝐶 = 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1331   ∈ wcel 1480  ∃wrex 2417   ↦ cmpt 3989  ran crn 4540 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-mpt 3991  df-cnv 4547  df-dm 4549  df-rn 4550 This theorem is referenced by: (None)
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