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Theorem dfdmf 4471
 Description: Definition of domain, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
dfdmf.1 xA
dfdmf.2 yA
Assertion
Ref Expression
dfdmf dom A = {xy xAy}
Distinct variable group:   x,y
Allowed substitution hints:   A(x,y)

Proof of Theorem dfdmf
Dummy variables w v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-dm 4298 . 2 dom A = {wv wAv}
2 nfcv 2175 . . . . 5 yw
3 dfdmf.2 . . . . 5 yA
4 nfcv 2175 . . . . 5 yv
52, 3, 4nfbr 3799 . . . 4 y wAv
6 nfv 1418 . . . 4 v wAy
7 breq2 3759 . . . 4 (v = y → (wAvwAy))
85, 6, 7cbvex 1636 . . 3 (v wAvy wAy)
98abbii 2150 . 2 {wv wAv} = {wy wAy}
10 nfcv 2175 . . . . 5 xw
11 dfdmf.1 . . . . 5 xA
12 nfcv 2175 . . . . 5 xy
1310, 11, 12nfbr 3799 . . . 4 x wAy
1413nfex 1525 . . 3 xy wAy
15 nfv 1418 . . 3 wy xAy
16 breq1 3758 . . . 4 (w = x → (wAyxAy))
1716exbidv 1703 . . 3 (w = x → (y wAyy xAy))
1814, 15, 17cbvab 2157 . 2 {wy wAy} = {xy xAy}
191, 9, 183eqtri 2061 1 dom A = {xy xAy}
 Colors of variables: wff set class Syntax hints:   = wceq 1242  ∃wex 1378  {cab 2023  Ⅎwnfc 2162   class class class wbr 3755  dom cdm 4288 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-dm 4298 This theorem is referenced by:  dmopab  4489
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