Theorem fnoprabg 3997

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 1404	. . . . . . 7 |- (E!zps -> E*zps)
2	1	imim2i 17	. . . . . 6 |- ((ph -> E!zps) -> (ph -> E*zps))
3		moanimv 1422	. . . . . 6 |- (E*z(ph /\ ps) <-> (ph -> E*zps))
4	2, 3	sylibr 200	. . . . 5 |- ((ph -> E!zps) -> E*z(ph /\ ps))
5	4	19.20i2 990	. . . 4 |- (A.xA.y(ph -> E!zps) -> A.xA.yE*z(ph /\ ps))
6		funoprabg 3995	. . . 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 1000	. . . . 5 |- (A.xA.y(ph -> E!zps) -> A.xA.xA.y(ph -> E!zps))
9		hba2 1010	. . . . 5 |- (A.xA.y(ph -> E!zps) -> A.yA.xA.y(ph -> E!zps))
10		pm3.26 319	. . . . . . . . 9 |- ((ph /\ ps) -> ph)
11	10	19.23aiv 1290	. . . . . . . 8 |- (E.z(ph /\ ps) -> ph)
12		euex 1387	. . . . . . . . . . 11 |- (E!zps -> E.zps)
13	12	imim2i 17	. . . . . . . . . 10 |- ((ph -> E!zps) -> (ph -> E.zps))
14	13	ancld 298	. . . . . . . . 9 |- ((ph -> E!zps) -> (ph -> (ph /\ E.zps)))
15		19.42v 1303	. . . . . . . . 9 |- (E.z(ph /\ ps) <-> (ph /\ E.zps))
16	14, 15	syl6ibr 213	. . . . . . . 8 |- ((ph -> E!zps) -> (ph -> E.z(ph /\ ps)))
17	11, 16	impbid2 516	. . . . . . 7 |- ((ph -> E!zps) -> (E.z(ph /\ ps) <-> ph))
18	17	a4s 981	. . . . . 6 |- (A.y(ph -> E!zps) -> (E.z(ph /\ ps) <-> ph))
19	18	a4s 981	. . . . 5 |- (A.xA.y(ph -> E!zps) -> (E.z(ph /\ ps) <-> ph))
20	8, 9, 19	opabbid 2659	. . . 4 |- (A.xA.y(ph -> E!zps) -> {<.x, y>. | E.z(ph /\ ps)} = {<.x, y>. | ph})
21		dmoprab 3987	. . . 4 |- dom {<.<.x, y>., z>. | (ph /\ ps)} = {<.x, y>. | E.z(ph /\ ps)}
22	20, 21	syl5eq 1511	. . 3 |- (A.xA.y(ph -> E!zps) -> dom {<.<.x, y>., z>. | (ph /\ ps)} = {<.x, y>. | ph})
23	7, 22	jca 288	. 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 3183	. 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 200	1 |- (A.xA.y(ph -> E!zps) -> {<.<.x, y>., z>. | (ph /\ ps)} Fn {<.x, y>. | ph})

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 951   = wceq 953  E.wex 977  E!weu 1373  E*wmo 1374  {copab 2656  dom cdm 3160  Fun wfun 3166   Fn wfn 3167  {copab2 3949

This theorem is referenced by:  fnoprab 3998

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-fun 3182  df-fn 3183  df-oprab 3951

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