Theorem fnopabg 4546

Description: Functionality and domain of an ordered-pair class abstraction.

Hypothesis

Ref	Expression
fnopabg.1	|- F = {<.x, y>. | (x e. A /\ ph)}

Assertion

Ref	Expression
fnopabg	|- (A.x e. A E!yph <-> F Fn A)

Distinct variable group:   x,y,A

Proof of Theorem fnopabg

Step	Hyp	Ref	Expression
1		df-fn 4009	. . . 4 |- ({<.x, y>. | (x e. A /\ ph)} Fn A <-> (Fun {<.x, y>. | (x e. A /\ ph)} /\ dom {<.x, y>. | (x e. A /\ ph)} = A))
2		hbra1 2147	. . . . . 6 |- (A.x e. A E!yph -> A.xA.x e. A E!yph)
3		ra4 2155	. . . . . . 7 |- (A.x e. A E!yph -> (x e. A -> E!yph))
4		eumo 1807	. . . . . . . . 9 |- (E!yph -> E*yph)
5	4	imim2i 11	. . . . . . . 8 |- ((x e. A -> E!yph) -> (x e. A -> E*yph))
6		moanimv 1829	. . . . . . . 8 |- (E*y(x e. A /\ ph) <-> (x e. A -> E*yph))
7	5, 6	sylibr 217	. . . . . . 7 |- ((x e. A -> E!yph) -> E*y(x e. A /\ ph))
8	3, 7	syl 12	. . . . . 6 |- (A.x e. A E!yph -> E*y(x e. A /\ ph))
9	2, 8	19.21ai 1345	. . . . 5 |- (A.x e. A E!yph -> A.xE*y(x e. A /\ ph))
10		funopab 4455	. . . . 5 |- (Fun {<.x, y>. | (x e. A /\ ph)} <-> A.xE*y(x e. A /\ ph))
11	9, 10	sylibr 217	. . . 4 |- (A.x e. A E!yph -> Fun {<.x, y>. | (x e. A /\ ph)})
12		euex 1788	. . . . . 6 |- (E!yph -> E.yph)
13	12	ralimi 2168	. . . . 5 |- (A.x e. A E!yph -> A.x e. A E.yph)
14		dmopab3 4169	. . . . 5 |- (A.x e. A E.yph <-> dom {<.x, y>. | (x e. A /\ ph)} = A)
15	13, 14	sylib 215	. . . 4 |- (A.x e. A E!yph -> dom {<.x, y>. | (x e. A /\ ph)} = A)
16	1, 11, 15	sylanbrc 527	. . 3 |- (A.x e. A E!yph -> {<.x, y>. | (x e. A /\ ph)} Fn A)
17		fnopabg.1	. . . 4 |- F = {<.x, y>. | (x e. A /\ ph)}
18	17	fneq1i 4507	. . 3 |- (F Fn A <-> {<.x, y>. | (x e. A /\ ph)} Fn A)
19	16, 18	sylibr 217	. 2 |- (A.x e. A E!yph -> F Fn A)
20		hbopab1 3562	. . . . 5 |- (z e. {<.x, y>. | (x e. A /\ ph)} -> A.x z e. {<.x, y>. | (x e. A /\ ph)})
21	17, 20	hbxfr 1992	. . . 4 |- (z e. F -> A.x z e. F)
22		ax-17 1317	. . . 4 |- (z e. A -> A.x z e. A)
23	21, 22	hbfn 4509	. . 3 |- (F Fn A -> A.x F Fn A)
24		fneu2 4519	. . . . . 6 |- ((F Fn A /\ x e. A) -> E!z<.x, z>. e. F)
25		ax-17 1317	. . . . . . . 8 |- (w e. <.x, z>. -> A.y w e. <.x, z>.)
26		hbopab2 3563	. . . . . . . . 9 |- (z e. {<.x, y>. | (x e. A /\ ph)} -> A.y z e. {<.x, y>. | (x e. A /\ ph)})
27	17, 26	hbxfr 1992	. . . . . . . 8 |- (z e. F -> A.y z e. F)
28	25, 27	hbel 1996	. . . . . . 7 |- (<.x, z>. e. F -> A.y<.x, z>. e. F)
29		ax-17 1317	. . . . . . 7 |- (<.x, y>. e. F -> A.z<.x, y>. e. F)
30		opeq2 3159	. . . . . . . 8 |- (z = y -> <.x, z>. = <.x, y>.)
31	30	eleq1d 1963	. . . . . . 7 |- (z = y -> (<.x, z>. e. F <-> <.x, y>. e. F))
32	28, 29, 31	cbveu 1785	. . . . . 6 |- (E!z<.x, z>. e. F <-> E!y<.x, y>. e. F)
33	24, 32	sylib 215	. . . . 5 |- ((F Fn A /\ x e. A) -> E!y<.x, y>. e. F)
34	17	eleq2i 1961	. . . . . . . . 9 |- (<.x, y>. e. F <-> <.x, y>. e. {<.x, y>. | (x e. A /\ ph)})
35		opabid 3557	. . . . . . . . 9 |- (<.x, y>. e. {<.x, y>. | (x e. A /\ ph)} <-> (x e. A /\ ph))
36	34, 35	bitri 190	. . . . . . . 8 |- (<.x, y>. e. F <-> (x e. A /\ ph))
37	36	eubii 1780	. . . . . . 7 |- (E!y<.x, y>. e. F <-> E!y(x e. A /\ ph))
38		euanv 1832	. . . . . . 7 |- (E!y(x e. A /\ ph) <-> (x e. A /\ E!yph))
39	37, 38	bitri 190	. . . . . 6 |- (E!y<.x, y>. e. F <-> (x e. A /\ E!yph))
40	39	simprbi 353	. . . . 5 |- (E!y<.x, y>. e. F -> E!yph)
41	33, 40	syl 12	. . . 4 |- ((F Fn A /\ x e. A) -> E!yph)
42	41	ex 402	. . 3 |- (F Fn A -> (x e. A -> E!yph))
43	23, 42	r19.21ai 2174	. 2 |- (F Fn A -> A.x e. A E!yph)
44	19, 43	impbii 174	1 |- (A.x e. A E!yph <-> F Fn A)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  E!weu 1771  E*wmo 1772  A.wral 2105  <.cop 3046  {copab 3395  dom cdm 3986  Fun wfun 3992   Fn wfn 3993

This theorem is referenced by:  fnopab2g 4547  fnopab 4548  elrnopabg 4773  fopab2 4796  en2d 5459  fopab2g 14485

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-ral 2109  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-rn 4005  df-fun 4008  df-fn 4009

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