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Theorem elrnmptf 39681
Description: The range of a function in maps-to notation. Same as elrnmpt 5404, 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 nfcv 2793 . . . 4 𝑥𝑦
2 elrnmptf.1 . . . 4 𝑥𝐶
31, 2nfeq 2805 . . 3 𝑥 𝑦 = 𝐶
4 eqeq1 2655 . . 3 (𝑦 = 𝐶 → (𝑦 = 𝐵𝐶 = 𝐵))
53, 4rexbid 3080 . 2 (𝑦 = 𝐶 → (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
6 elrnmptf.2 . . 3 𝐹 = (𝑥𝐴𝐵)
76rnmpt 5403 . 2 ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
85, 7elab2g 3385 1 (𝐶𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1523  wcel 2030  wnfc 2780  wrex 2942  cmpt 4762  ran crn 5144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-mpt 4763  df-cnv 5151  df-dm 5153  df-rn 5154
This theorem is referenced by:  elrnmpt1sf  39690
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