Theorem fnoprabg 5585

Description: Functionality and domain of an operation class abstraction. (Contributed by set.mm contributors, 28-Aug-2007.)

Assertion

Ref	Expression
fnoprabg	|- (A.xA.y(ph -> E!zps) -> {<.<.x, y>., z>. | (ph /\ ps)} Fn {<.x, y>. | ph})

Distinct variable groups:   x,y,z   ph,z

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

Proof of Theorem fnoprabg

Step	Hyp	Ref	Expression
1		eumo 2244	. . . . . 6 |- (E!zps -> E*zps)
2	1	imim2i 13	. . . . 5 |- ((ph -> E!zps) -> (ph -> E*zps))
3		moanimv 2262	. . . . 5 |- (E*z(ph /\ ps) <-> (ph -> E*zps))
4	2, 3	sylibr 203	. . . 4 |- ((ph -> E!zps) -> E*z(ph /\ ps))
5	4	2alimi 1560	. . 3 |- (A.xA.y(ph -> E!zps) -> A.xA.yE*z(ph /\ ps))
6		funoprabg 5583	. . 3 |- (A.xA.yE*z(ph /\ ps) -> Fun {<.<.x, y>., z>. | (ph /\ ps)})
7	5, 6	syl 15	. 2 |- (A.xA.y(ph -> E!zps) -> Fun {<.<.x, y>., z>. | (ph /\ ps)})
8		dmoprab 5574	. . 3 |- dom {<.<.x, y>., z>. | (ph /\ ps)} = {<.x, y>. | E.z(ph /\ ps)}
9		nfa1 1788	. . . 4 |- F/xA.xA.y(ph -> E!zps)
10		nfa2 1855	. . . 4 |- F/yA.xA.y(ph -> E!zps)
11		simpl 443	. . . . . . . 8 |- ((ph /\ ps) -> ph)
12	11	exlimiv 1634	. . . . . . 7 |- (E.z(ph /\ ps) -> ph)
13		euex 2227	. . . . . . . . . 10 |- (E!zps -> E.zps)
14	13	imim2i 13	. . . . . . . . 9 |- ((ph -> E!zps) -> (ph -> E.zps))
15	14	ancld 536	. . . . . . . 8 |- ((ph -> E!zps) -> (ph -> (ph /\ E.zps)))
16		19.42v 1905	. . . . . . . 8 |- (E.z(ph /\ ps) <-> (ph /\ E.zps))
17	15, 16	syl6ibr 218	. . . . . . 7 |- ((ph -> E!zps) -> (ph -> E.z(ph /\ ps)))
18	12, 17	impbid2 195	. . . . . 6 |- ((ph -> E!zps) -> (E.z(ph /\ ps) <-> ph))
19	18	sps 1754	. . . . 5 |- (A.y(ph -> E!zps) -> (E.z(ph /\ ps) <-> ph))
20	19	sps 1754	. . . 4 |- (A.xA.y(ph -> E!zps) -> (E.z(ph /\ ps) <-> ph))
21	9, 10, 20	opabbid 4624	. . 3 |- (A.xA.y(ph -> E!zps) -> {<.x, y>. | E.z(ph /\ ps)} = {<.x, y>. | ph})
22	8, 21	syl5eq 2397	. 2 |- (A.xA.y(ph -> E!zps) -> dom {<.<.x, y>., z>. | (ph /\ ps)} = {<.x, y>. | ph})
23		df-fn 4790	. 2 |- ({<.<.x, y>., z>. | (ph /\ ps)} Fn {<.x, y>. | ph} <-> (Fun {<.<.x, y>., z>. | (ph /\ ps)} /\ dom {<.<.x, y>., z>. | (ph /\ ps)} = {<.x, y>. | ph}))
24	7, 22, 23	sylanbrc 645	1 |- (A.xA.y(ph -> E!zps) -> {<.<.x, y>., z>. | (ph /\ ps)} Fn {<.x, y>. | ph})

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541   = wceq 1642  E!weu 2204  E*wmo 2205  {copab 4622  dom cdm 4772  Fun wfun 4775   Fn wfn 4776  {coprab 5527

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 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-co 4726  df-ima 4727  df-id 4767  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-oprab 5528

This theorem is referenced by:  fnoprab  5586  ovg  5601