Theorem fnoprabg 4003

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 1409	. . . . . . 7 |- (E!zps -> E*zps)
2	1	imim2i 17	. . . . . 6 |- ((ph -> E!zps) -> (ph -> E*zps))
3		moanimv 1427	. . . . . 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 991	. . . 4 |- (A.xA.y(ph -> E!zps) -> A.xA.yE*z(ph /\ ps))
6		funoprabg 4001	. . . 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 1001	. . . . 5 |- (A.xA.y(ph -> E!zps) -> A.xA.xA.y(ph -> E!zps))
9		hba2 1011	. . . . 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 1293	. . . . . . . 8 |- (E.z(ph /\ ps) -> ph)
12		euex 1392	. . . . . . . . . . 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 1306	. . . . . . . . 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 517	. . . . . . 7 |- ((ph -> E!zps) -> (E.z(ph /\ ps) <-> ph))
18	17	a4s 982	. . . . . 6 |- (A.y(ph -> E!zps) -> (E.z(ph /\ ps) <-> ph))
19	18	a4s 982	. . . . 5 |- (A.xA.y(ph -> E!zps) -> (E.z(ph /\ ps) <-> ph))
20	8, 9, 19	opabbid 2664	. . . 4 |- (A.xA.y(ph -> E!zps) -> {<.x, y>. | E.z(ph /\ ps)} = {<.x, y>. | ph})
21		dmoprab 3993	. . . 4 |- dom {<.<.x, y>., z>. | (ph /\ ps)} = {<.x, y>. | E.z(ph /\ ps)}
22	20, 21	syl5eq 1516	. . 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 3188	. 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 952   = wceq 954  E.wex 978  E!weu 1378  E*wmo 1379  {copab 2661  dom cdm 3165  Fun wfun 3171   Fn wfn 3172  {copab2 3955

This theorem is referenced by:  fnoprab 4004

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-fun 3187  df-fn 3188  df-oprab 3957

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