Theorem fnoprabg 4089

Description: Functionality and domain of an operation class abstraction.

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

Proof of Theorem fnoprabg

Step	Hyp	Ref	Expression
1		eumo 1444	. . . . . . 7 |- (E!zps -> E*zps)
2	1	imim2i 17	. . . . . 6 |- ((ph -> E!zps) -> (ph -> E*zps))
3		moanimv 1462	. . . . . 6 |- (E*z(ph /\ ps) <-> (ph -> E*zps))
4	2, 3	sylibr 198	. . . . 5 |- ((ph -> E!zps) -> E*z(ph /\ ps))
5	4	19.20i2 1025	. . . 4 |- (A.xA.y(ph -> E!zps) -> A.xA.yE*z(ph /\ ps))
6		funoprabg 4087	. . . 4 |- (A.xA.yE*z(ph /\ ps) -> Fun {<.<.x, y>., z>. | (ph /\ ps)})
7	5, 6	syl 10	. . 3 |- (A.xA.y(ph -> E!zps) -> Fun {<.<.x, y>., z>. | (ph /\ ps)})
8		hba1 1035	. . . . 5 |- (A.xA.y(ph -> E!zps) -> A.xA.xA.y(ph -> E!zps))
9		hba2 1045	. . . . 5 |- (A.xA.y(ph -> E!zps) -> A.yA.xA.y(ph -> E!zps))
10		pm3.26 317	. . . . . . . . 9 |- ((ph /\ ps) -> ph)
11	10	19.23aiv 1328	. . . . . . . 8 |- (E.z(ph /\ ps) -> ph)
12		euex 1427	. . . . . . . . . . 11 |- (E!zps -> E.zps)
13	12	imim2i 17	. . . . . . . . . 10 |- ((ph -> E!zps) -> (ph -> E.zps))
14	13	ancld 296	. . . . . . . . 9 |- ((ph -> E!zps) -> (ph -> (ph /\ E.zps)))
15		19.42v 1341	. . . . . . . . 9 |- (E.z(ph /\ ps) <-> (ph /\ E.zps))
16	14, 15	syl6ibr 211	. . . . . . . 8 |- ((ph -> E!zps) -> (ph -> E.z(ph /\ ps)))
17	11, 16	impbid2 520	. . . . . . 7 |- ((ph -> E!zps) -> (E.z(ph /\ ps) <-> ph))
18	17	a4s 1016	. . . . . 6 |- (A.y(ph -> E!zps) -> (E.z(ph /\ ps) <-> ph))
19	18	a4s 1016	. . . . 5 |- (A.xA.y(ph -> E!zps) -> (E.z(ph /\ ps) <-> ph))
20	8, 9, 19	opabbid 2720	. . . 4 |- (A.xA.y(ph -> E!zps) -> {<.x, y>. | E.z(ph /\ ps)} = {<.x, y>. | ph})
21		dmoprab 4079	. . . 4 |- dom {<.<.x, y>., z>. | (ph /\ ps)} = {<.x, y>. | E.z(ph /\ ps)}
22	20, 21	syl5eq 1556	. . 3 |- (A.xA.y(ph -> E!zps) -> dom {<.<.x, y>., z>. | (ph /\ ps)} = {<.x, y>. | ph})
23	7, 22	jca 286	. 2 |- (A.xA.y(ph -> E!zps) -> (Fun {<.<.x, y>., z>. | (ph /\ ps)} /\ dom {<.<.x, y>., z>. | (ph /\ ps)} = {<.x, y>. | ph}))
24		df-fn 3248	. 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}))
25	23, 24	sylibr 198	1 |- (A.xA.y(ph -> E!zps) -> {<.<.x, y>., z>. | (ph /\ ps)} Fn {<.x, y>. | ph})

Syntax hints:   -> wi 3   <-> wb 144   /\ wa 221  A.wal 986   = wceq 988  E.wex 1012  E!weu 1413  E*wmo 1414  {copab 2717  dom cdm 3225  Fun wfun 3231   Fn wfn 3232  {copab2 4040

This theorem is referenced by:  fnoprab 4090

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 994  ax-gen 995  ax-8 996  ax-9 997  ax-10 998  ax-11 999  ax-12 1000  ax-13 1001  ax-14 1002  ax-17 1003  ax-4 1005  ax-5o 1007  ax-6o 1010  ax-9o 1155  ax-10o 1173  ax-16 1243  ax-11o 1251  ax-ext 1494  ax-sep 2754  ax-pow 2794  ax-pr 2832

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1013  df-sb 1205  df-eu 1415  df-mo 1416  df-clab 1500  df-cleq 1505  df-clel 1508  df-ne 1624  df-v 1850  df-dif 2093  df-un 2094  df-in 2095  df-ss 2097  df-nul 2325  df-pw 2447  df-sn 2457  df-pr 2458  df-op 2461  df-br 2670  df-opab 2718  df-id 2889  df-xp 3239  df-rel 3240  df-cnv 3241  df-co 3242  df-dm 3243  df-fun 3247  df-fn 3248  df-oprab 4042

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