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Theorem elrnmptf 41800
 Description: The range of a function in maps-to notation. Same as elrnmpt 5796, but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
elrnmptf.1 𝑥𝐶
elrnmptf.2 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
elrnmptf (𝐶𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))

Proof of Theorem elrnmptf
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elrnmptf.1 . . . 4 𝑥𝐶
21nfeq2 2975 . . 3 𝑥 𝑦 = 𝐶
3 eqeq1 2805 . . 3 (𝑦 = 𝐶 → (𝑦 = 𝐵𝐶 = 𝐵))
42, 3rexbid 3282 . 2 (𝑦 = 𝐶 → (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
5 elrnmptf.2 . . 3 𝐹 = (𝑥𝐴𝐵)
65rnmpt 5795 . 2 ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
74, 6elab2g 3619 1 (𝐶𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2112  Ⅎwnfc 2939  ∃wrex 3110   ↦ cmpt 5113  ran crn 5524 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-rex 3115  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-mpt 5114  df-cnv 5531  df-dm 5533  df-rn 5534 This theorem is referenced by:  elrnmpt1sf  41809
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