Theorem fnoprabg 4941

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		df-fn 4009	. 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}))
2		eumo 1807	. . . . . 6 |- (E!zps -> E*zps)
3	2	imim2i 11	. . . . 5 |- ((ph -> E!zps) -> (ph -> E*zps))
4		moanimv 1829	. . . . 5 |- (E*z(ph /\ ps) <-> (ph -> E*zps))
5	3, 4	sylibr 217	. . . 4 |- ((ph -> E!zps) -> E*z(ph /\ ps))
6	5	2alimi 1339	. . 3 |- (A.xA.y(ph -> E!zps) -> A.xA.yE*z(ph /\ ps))
7		funoprabg 4939	. . 3 |- (A.xA.yE*z(ph /\ ps) -> Fun {<.<.x, y>., z>. | (ph /\ ps)})
8	6, 7	syl 12	. 2 |- (A.xA.y(ph -> E!zps) -> Fun {<.<.x, y>., z>. | (ph /\ ps)})
9		hba1 1350	. . . 4 |- (A.xA.y(ph -> E!zps) -> A.xA.xA.y(ph -> E!zps))
10		hba2 1360	. . . 4 |- (A.xA.y(ph -> E!zps) -> A.yA.xA.y(ph -> E!zps))
11		simpl 346	. . . . . . . 8 |- ((ph /\ ps) -> ph)
12	11	19.23aiv 1674	. . . . . . 7 |- (E.z(ph /\ ps) -> ph)
13		euex 1788	. . . . . . . . . 10 |- (E!zps -> E.zps)
14	13	imim2i 11	. . . . . . . . 9 |- ((ph -> E!zps) -> (ph -> E.zps))
15	14	ancld 322	. . . . . . . 8 |- ((ph -> E!zps) -> (ph -> (ph /\ E.zps)))
16		19.42v 1688	. . . . . . . 8 |- (E.z(ph /\ ps) <-> (ph /\ E.zps))
17	15, 16	syl6ibr 230	. . . . . . 7 |- ((ph -> E!zps) -> (ph -> E.z(ph /\ ps)))
18	12, 17	impbid2 576	. . . . . 6 |- ((ph -> E!zps) -> (E.z(ph /\ ps) <-> ph))
19	18	a4s 1330	. . . . 5 |- (A.y(ph -> E!zps) -> (E.z(ph /\ ps) <-> ph))
20	19	a4s 1330	. . . 4 |- (A.xA.y(ph -> E!zps) -> (E.z(ph /\ ps) <-> ph))
21	9, 10, 20	opabbid 3399	. . 3 |- (A.xA.y(ph -> E!zps) -> {<.x, y>. | E.z(ph /\ ps)} = {<.x, y>. | ph})
22		dmoprab 4931	. . 3 |- dom {<.<.x, y>., z>. | (ph /\ ps)} = {<.x, y>. | E.z(ph /\ ps)}
23	21, 22	syl5eq 1940	. 2 |- (A.xA.y(ph -> E!zps) -> dom {<.<.x, y>., z>. | (ph /\ ps)} = {<.x, y>. | ph})
24	1, 8, 23	sylanbrc 527	1 |- (A.xA.y(ph -> E!zps) -> {<.<.x, y>., z>. | (ph /\ ps)} Fn {<.x, y>. | ph})

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298  E.wex 1326  E!weu 1771  E*wmo 1772  {copab 3395  dom cdm 3986  Fun wfun 3992   Fn wfn 3993  {copab2 4885

This theorem is referenced by:  fnoprab 4942  hartog 5693  hartogOLD 15384  oprabvalg 15706  eroprf 15735

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-fun 4008  df-fn 4009  df-oprab 4887

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