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Theorem fvelrnbf 38200
 Description: A version of fvelrnb 6153 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
fvelrnbf.1 𝑥𝐴
fvelrnbf.2 𝑥𝐵
fvelrnbf.3 𝑥𝐹
Assertion
Ref Expression
fvelrnbf (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥𝐴 (𝐹𝑥) = 𝐵))

Proof of Theorem fvelrnbf
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fvelrnb 6153 . 2 (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 ↔ ∃𝑦𝐴 (𝐹𝑦) = 𝐵))
2 nfcv 2751 . . 3 𝑦𝐴
3 fvelrnbf.1 . . 3 𝑥𝐴
4 fvelrnbf.3 . . . . 5 𝑥𝐹
5 nfcv 2751 . . . . 5 𝑥𝑦
64, 5nffv 6110 . . . 4 𝑥(𝐹𝑦)
7 fvelrnbf.2 . . . 4 𝑥𝐵
86, 7nfeq 2762 . . 3 𝑥(𝐹𝑦) = 𝐵
9 nfv 1830 . . 3 𝑦(𝐹𝑥) = 𝐵
10 fveq2 6103 . . . 4 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
1110eqeq1d 2612 . . 3 (𝑦 = 𝑥 → ((𝐹𝑦) = 𝐵 ↔ (𝐹𝑥) = 𝐵))
122, 3, 8, 9, 11cbvrexf 3142 . 2 (∃𝑦𝐴 (𝐹𝑦) = 𝐵 ↔ ∃𝑥𝐴 (𝐹𝑥) = 𝐵)
131, 12syl6bb 275 1 (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥𝐴 (𝐹𝑥) = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  Ⅎwnfc 2738  ∃wrex 2897  ran crn 5039   Fn wfn 5799  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812 This theorem is referenced by:  refsumcn  38212  stoweidlem29  38922
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