Theorem fnoprabg 4012

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 1411	. . . . . . 7 |- (E!zps -> E*zps)
2	1	imim2i 17	. . . . . 6 |- ((ph -> E!zps) -> (ph -> E*zps))
3		moanimv 1429	. . . . . 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 993	. . . 4 |- (A.xA.y(ph -> E!zps) -> A.xA.yE*z(ph /\ ps))
6		funoprabg 4010	. . . 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 1003	. . . . 5 |- (A.xA.y(ph -> E!zps) -> A.xA.xA.y(ph -> E!zps))
9		hba2 1013	. . . . 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 1295	. . . . . . . 8 |- (E.z(ph /\ ps) -> ph)
12		euex 1394	. . . . . . . . . . 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 1308	. . . . . . . . 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 518	. . . . . . 7 |- ((ph -> E!zps) -> (E.z(ph /\ ps) <-> ph))
18	17	a4s 984	. . . . . 6 |- (A.y(ph -> E!zps) -> (E.z(ph /\ ps) <-> ph))
19	18	a4s 984	. . . . 5 |- (A.xA.y(ph -> E!zps) -> (E.z(ph /\ ps) <-> ph))
20	8, 9, 19	opabbid 2669	. . . 4 |- (A.xA.y(ph -> E!zps) -> {<.x, y>. | E.z(ph /\ ps)} = {<.x, y>. | ph})
21		dmoprab 4002	. . . 4 |- dom {<.<.x, y>., z>. | (ph /\ ps)} = {<.x, y>. | E.z(ph /\ ps)}
22	20, 21	syl5eq 1519	. . 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 3193	. 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 954   = wceq 956  E.wex 980  E!weu 1380  E*wmo 1381  {copab 2666  dom cdm 3170  Fun wfun 3176   Fn wfn 3177  {copab2 3964

This theorem is referenced by:  fnoprab 4013

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-fun 3192  df-fn 3193  df-oprab 3966

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