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Theorem dfrnf 5906
Description: Definition of range, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
dfrnf.1 𝑥𝐴
dfrnf.2 𝑦𝐴
Assertion
Ref Expression
dfrnf ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem dfrnf
Dummy variables 𝑤 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfrn2 5845 . 2 ran 𝐴 = {𝑤 ∣ ∃𝑣 𝑣𝐴𝑤}
2 nfcv 2904 . . . . 5 𝑥𝑣
3 dfrnf.1 . . . . 5 𝑥𝐴
4 nfcv 2904 . . . . 5 𝑥𝑤
52, 3, 4nfbr 5153 . . . 4 𝑥 𝑣𝐴𝑤
6 nfv 1918 . . . 4 𝑣 𝑥𝐴𝑤
7 breq1 5109 . . . 4 (𝑣 = 𝑥 → (𝑣𝐴𝑤𝑥𝐴𝑤))
85, 6, 7cbvexv1 2339 . . 3 (∃𝑣 𝑣𝐴𝑤 ↔ ∃𝑥 𝑥𝐴𝑤)
98abbii 2803 . 2 {𝑤 ∣ ∃𝑣 𝑣𝐴𝑤} = {𝑤 ∣ ∃𝑥 𝑥𝐴𝑤}
10 nfcv 2904 . . . . 5 𝑦𝑥
11 dfrnf.2 . . . . 5 𝑦𝐴
12 nfcv 2904 . . . . 5 𝑦𝑤
1310, 11, 12nfbr 5153 . . . 4 𝑦 𝑥𝐴𝑤
1413nfex 2318 . . 3 𝑦𝑥 𝑥𝐴𝑤
15 nfv 1918 . . 3 𝑤𝑥 𝑥𝐴𝑦
16 breq2 5110 . . . 4 (𝑤 = 𝑦 → (𝑥𝐴𝑤𝑥𝐴𝑦))
1716exbidv 1925 . . 3 (𝑤 = 𝑦 → (∃𝑥 𝑥𝐴𝑤 ↔ ∃𝑥 𝑥𝐴𝑦))
1814, 15, 17cbvabw 2807 . 2 {𝑤 ∣ ∃𝑥 𝑥𝐴𝑤} = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
191, 9, 183eqtri 2765 1 ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wex 1782  {cab 2710  wnfc 2884   class class class wbr 5106  ran crn 5635
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 5257  ax-nul 5264  ax-pr 5385
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-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-cnv 5642  df-dm 5644  df-rn 5645
This theorem is referenced by:  rnopab  5910
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