Theorem dmoprab 5575

Description: The domain of an operation class abstraction. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by set.mm contributors, 17-Mar-1995.) (Revised by set.mm contributors, 24-Jul-2012.)

Assertion

Ref	Expression
dmoprab	|- dom {<.<.x, y>., z>. | ph} = {<.x, y>. | E.zph}

Distinct variable groups:   x,z   y,z

Allowed substitution hints:   ph(x,y,z)

Proof of Theorem dmoprab

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfoprab2 5559	. . 3 |- {<.<.x, y>., z>. | ph} = {<.w, z>. | E.xE.y(w = <.x, y>. /\ ph)}
2	1	dmeqi 4909	. 2 |- dom {<.<.x, y>., z>. | ph} = dom {<.w, z>. | E.xE.y(w = <.x, y>. /\ ph)}
3		dmopab 4916	. 2 |- dom {<.w, z>. | E.xE.y(w = <.x, y>. /\ ph)} = {w | E.zE.xE.y(w = <.x, y>. /\ ph)}
4		exrot3 1744	. . . . 5 |- (E.zE.xE.y(w = <.x, y>. /\ ph) <-> E.xE.yE.z(w = <.x, y>. /\ ph))
5		19.42v 1905	. . . . . 6 |- (E.z(w = <.x, y>. /\ ph) <-> (w = <.x, y>. /\ E.zph))
6	5	2exbii 1583	. . . . 5 |- (E.xE.yE.z(w = <.x, y>. /\ ph) <-> E.xE.y(w = <.x, y>. /\ E.zph))
7	4, 6	bitri 240	. . . 4 |- (E.zE.xE.y(w = <.x, y>. /\ ph) <-> E.xE.y(w = <.x, y>. /\ E.zph))
8	7	abbii 2466	. . 3 |- {w | E.zE.xE.y(w = <.x, y>. /\ ph)} = {w | E.xE.y(w = <.x, y>. /\ E.zph)}
9		df-opab 4624	. . 3 |- {<.x, y>. | E.zph} = {w | E.xE.y(w = <.x, y>. /\ E.zph)}
10	8, 9	eqtr4i 2376	. 2 |- {w | E.zE.xE.y(w = <.x, y>. /\ ph)} = {<.x, y>. | E.zph}
11	2, 3, 10	3eqtri 2377	1 |- dom {<.<.x, y>., z>. | ph} = {<.x, y>. | E.zph}

Syntax hints:   /\ wa 358  E.wex 1541   = wceq 1642  {cab 2339  <.cop 4562  {copab 4623  dom cdm 4773  {coprab 5528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-ima 4728  df-cnv 4786  df-rn 4787  df-dm 4788  df-oprab 5529

This theorem is referenced by:  dmoprabss  5576  fnoprabg  5586