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Theorem elrnmptf 40176
 Description: The range of a function in maps-to notation. Same as elrnmpt 5606, 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 2970 . . . 4 𝑥𝑦
2 elrnmptf.1 . . . 4 𝑥𝐶
31, 2nfeq 2982 . . 3 𝑥 𝑦 = 𝐶
4 eqeq1 2830 . . 3 (𝑦 = 𝐶 → (𝑦 = 𝐵𝐶 = 𝐵))
53, 4rexbid 3262 . 2 (𝑦 = 𝐶 → (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
6 elrnmptf.2 . . 3 𝐹 = (𝑥𝐴𝐵)
76rnmpt 5605 . 2 ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
85, 7elab2g 3575 1 (𝐶𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   = wceq 1658   ∈ wcel 2166  Ⅎwnfc 2957  ∃wrex 3119   ↦ cmpt 4953  ran crn 5344 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pr 5128 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-rex 3124  df-rab 3127  df-v 3417  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4875  df-opab 4937  df-mpt 4954  df-cnv 5351  df-dm 5353  df-rn 5354 This theorem is referenced by:  elrnmpt1sf  40185
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