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Theorem elrnmptf 43749
Description: The range of a function in maps-to notation. Same as elrnmpt 5950, 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 2921 . . 3 𝑥 𝑦 = 𝐶
3 eqeq1 2737 . . 3 (𝑦 = 𝐶 → (𝑦 = 𝐵𝐶 = 𝐵))
42, 3rexbid 3272 . 2 (𝑦 = 𝐶 → (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
5 elrnmptf.2 . . 3 𝐹 = (𝑥𝐴𝐵)
65rnmpt 5949 . 2 ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
74, 6elab2g 3668 1 (𝐶𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wcel 2107  wnfc 2884  wrex 3071  cmpt 5227  ran crn 5673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5145  df-opab 5207  df-mpt 5228  df-cnv 5680  df-dm 5682  df-rn 5683
This theorem is referenced by:  elrnmpt1sf  43758
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