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Theorem elrnmptf 38165
Description: The range of a function in maps-to notation. Same as elrnmpt 5280, 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 2750 . . . 4 𝑥𝑦
2 elrnmptf.1 . . . 4 𝑥𝐶
31, 2nfeq 2761 . . 3 𝑥 𝑦 = 𝐶
4 eqeq1 2613 . . 3 (𝑦 = 𝐶 → (𝑦 = 𝐵𝐶 = 𝐵))
53, 4rexbid 3032 . 2 (𝑦 = 𝐶 → (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
6 elrnmptf.2 . . 3 𝐹 = (𝑥𝐴𝐵)
76rnmpt 5279 . 2 ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
85, 7elab2g 3321 1 (𝐶𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194   = wceq 1474  wcel 1976  wnfc 2737  wrex 2896  cmpt 4637  ran crn 5029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-mpt 4639  df-cnv 5036  df-dm 5038  df-rn 5039
This theorem is referenced by:  elrnmpt1sf  38174
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