Theorem fnopabg 3690

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		hbra1 1725	. . . . . . 7 |- (A.x e. A E!yph -> A.xA.x e. A E!yph)
2		ra4 1732	. . . . . . . 8 |- (A.x e. A E!yph -> (x e. A -> E!yph))
3		eumo 1444	. . . . . . . . . 10 |- (E!yph -> E*yph)
4	3	imim2i 17	. . . . . . . . 9 |- ((x e. A -> E!yph) -> (x e. A -> E*yph))
5		moanimv 1462	. . . . . . . . 9 |- (E*y(x e. A /\ ph) <-> (x e. A -> E*yph))
6	4, 5	sylibr 198	. . . . . . . 8 |- ((x e. A -> E!yph) -> E*y(x e. A /\ ph))
7	2, 6	syl 10	. . . . . . 7 |- (A.x e. A E!yph -> E*y(x e. A /\ ph))
8	1, 7	19.21ai 1030	. . . . . 6 |- (A.x e. A E!yph -> A.xE*y(x e. A /\ ph))
9		funopab 3623	. . . . . 6 |- (Fun {<.x, y>. | (x e. A /\ ph)} <-> A.xE*y(x e. A /\ ph))
10	8, 9	sylibr 198	. . . . 5 |- (A.x e. A E!yph -> Fun {<.x, y>. | (x e. A /\ ph)})
11		euex 1427	. . . . . . 7 |- (E!yph -> E.yph)
12	11	r19.20si 1744	. . . . . 6 |- (A.x e. A E!yph -> A.x e. A E.yph)
13		dmopab3 3386	. . . . . 6 |- (A.x e. A E.yph <-> dom {<.x, y>. | (x e. A /\ ph)} = A)
14	12, 13	sylib 196	. . . . 5 |- (A.x e. A E!yph -> dom {<.x, y>. | (x e. A /\ ph)} = A)
15	10, 14	jca 286	. . . 4 |- (A.x e. A E!yph -> (Fun {<.x, y>. | (x e. A /\ ph)} /\ dom {<.x, y>. | (x e. A /\ ph)} = A))
16		df-fn 3248	. . . 4 |- ({<.x, y>. | (x e. A /\ ph)} Fn A <-> (Fun {<.x, y>. | (x e. A /\ ph)} /\ dom {<.x, y>. | (x e. A /\ ph)} = A))
17	15, 16	sylibr 198	. . 3 |- (A.x e. A E!yph -> {<.x, y>. | (x e. A /\ ph)} Fn A)
18		fnopabg.1	. . . 4 |- F = {<.x, y>. | (x e. A /\ ph)}
19		fneq1 3657	. . . 4 |- (F = {<.x, y>. | (x e. A /\ ph)} -> (F Fn A <-> {<.x, y>. | (x e. A /\ ph)} Fn A))
20	18, 19	ax-mp 7	. . 3 |- (F Fn A <-> {<.x, y>. | (x e. A /\ ph)} Fn A)
21	17, 20	sylibr 198	. 2 |- (A.x e. A E!yph -> F Fn A)
22		hbopab1 2866	. . . . 5 |- (z e. {<.x, y>. | (x e. A /\ ph)} -> A.x z e. {<.x, y>. | (x e. A /\ ph)})
23	18, 22	hbxfr 1600	. . . 4 |- (z e. F -> A.x z e. F)
24		ax-17 1003	. . . 4 |- (z e. A -> A.x z e. A)
25	23, 24	hbfn 3659	. . 3 |- (F Fn A -> A.x F Fn A)
26		fneu2 3668	. . . . . 6 |- ((F Fn A /\ x e. A) -> E!z<.x, z>. e. F)
27		ax-17 1003	. . . . . . . 8 |- (w e. <.x, z>. -> A.y w e. <.x, z>.)
28		hbopab2 2867	. . . . . . . . 9 |- (z e. {<.x, y>. | (x e. A /\ ph)} -> A.y z e. {<.x, y>. | (x e. A /\ ph)})
29	18, 28	hbxfr 1600	. . . . . . . 8 |- (z e. F -> A.y z e. F)
30	27, 29	hbel 1603	. . . . . . 7 |- (<.x, z>. e. F -> A.y<.x, z>. e. F)
31		ax-17 1003	. . . . . . 7 |- (<.x, y>. e. F -> A.z<.x, y>. e. F)
32		opeq2 2536	. . . . . . . 8 |- (z = y -> <.x, z>. = <.x, y>.)
33	32	eleq1d 1577	. . . . . . 7 |- (z = y -> (<.x, z>. e. F <-> <.x, y>. e. F))
34	30, 31, 33	cbveu 1424	. . . . . 6 |- (E!z<.x, z>. e. F <-> E!y<.x, y>. e. F)
35	26, 34	sylib 196	. . . . 5 |- ((F Fn A /\ x e. A) -> E!y<.x, y>. e. F)
36	18	eleq2i 1575	. . . . . . . . 9 |- (<.x, y>. e. F <-> <.x, y>. e. {<.x, y>. | (x e. A /\ ph)})
37		opabid 2863	. . . . . . . . 9 |- (<.x, y>. e. {<.x, y>. | (x e. A /\ ph)} <-> (x e. A /\ ph))
38	36, 37	bitri 171	. . . . . . . 8 |- (<.x, y>. e. F <-> (x e. A /\ ph))
39	38	eubii 1420	. . . . . . 7 |- (E!y<.x, y>. e. F <-> E!y(x e. A /\ ph))
40		euanv 1465	. . . . . . 7 |- (E!y(x e. A /\ ph) <-> (x e. A /\ E!yph))
41	39, 40	bitri 171	. . . . . 6 |- (E!y<.x, y>. e. F <-> (x e. A /\ E!yph))
42	41	pm3.27bi 324	. . . . 5 |- (E!y<.x, y>. e. F -> E!yph)
43	35, 42	syl 10	. . . 4 |- ((F Fn A /\ x e. A) -> E!yph)
44	43	ex 371	. . 3 |- (F Fn A -> (x e. A -> E!yph))
45	25, 44	r19.21ai 1750	. 2 |- (F Fn A -> A.x e. A E!yph)
46	21, 45	impbii 155	1 |- (A.x e. A E!yph <-> F Fn A)

Syntax hints:   -> wi 3   <-> wb 144   /\ wa 221  A.wal 986   = wceq 988   e. wcel 990  E.wex 1012  E!weu 1413  E*wmo 1414  A.wral 1683  <.cop 2456  {copab 2717  dom cdm 3225  Fun wfun 3231   Fn wfn 3232

This theorem is referenced by:  fnopab2g 3691  fnopab 3692  elrnopabg 3876  fopab2 3899  en2d 4487

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 994  ax-gen 995  ax-8 996  ax-9 997  ax-10 998  ax-11 999  ax-12 1000  ax-13 1001  ax-14 1002  ax-17 1003  ax-4 1005  ax-5o 1007  ax-6o 1010  ax-9o 1155  ax-10o 1173  ax-16 1243  ax-11o 1251  ax-ext 1494  ax-sep 2754  ax-pow 2794  ax-pr 2832

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1013  df-sb 1205  df-eu 1415  df-mo 1416  df-clab 1500  df-cleq 1505  df-clel 1508  df-ne 1624  df-ral 1687  df-v 1850  df-dif 2093  df-un 2094  df-in 2095  df-ss 2097  df-nul 2325  df-pw 2447  df-sn 2457  df-pr 2458  df-op 2461  df-br 2670  df-opab 2718  df-id 2889  df-xp 3239  df-rel 3240  df-cnv 3241  df-co 3242  df-dm 3243  df-rn 3244  df-fun 3247  df-fn 3248

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