New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  dfdmf GIF version

Theorem dfdmf 4905
 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 = {x y 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 dfdm2 4900 . 2 dom A = {w v wAv}
2 nfcv 2489 . . . . 5 yw
3 dfdmf.2 . . . . 5 yA
4 nfcv 2489 . . . . 5 yv
52, 3, 4nfbr 4683 . . . 4 y wAv
6 nfv 1619 . . . 4 v wAy
7 breq2 4643 . . . 4 (v = y → (wAvwAy))
85, 6, 7cbvex 1985 . . 3 (v wAvy wAy)
98abbii 2465 . 2 {w v wAv} = {w y wAy}
10 nfcv 2489 . . . . 5 xw
11 dfdmf.1 . . . . 5 xA
12 nfcv 2489 . . . . 5 xy
1310, 11, 12nfbr 4683 . . . 4 x wAy
1413nfex 1843 . . 3 xy wAy
15 nfv 1619 . . 3 wy xAy
16 breq1 4642 . . . 4 (w = x → (wAyxAy))
1716exbidv 1626 . . 3 (w = x → (y wAyy xAy))
1814, 15, 17cbvab 2471 . 2 {w y wAy} = {x y xAy}
191, 9, 183eqtri 2377 1 dom A = {x y xAy}
 Colors of variables: wff setvar class Syntax hints:  ∃wex 1541   = wceq 1642  {cab 2339  Ⅎwnfc 2476   class class class wbr 4639  dom cdm 4772 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-ima 4727  df-cnv 4785  df-rn 4786  df-dm 4787 This theorem is referenced by:  dmopab  4915
 Copyright terms: Public domain W3C validator