Theorem fnopabg 3555

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 1663	. . . . . . 7 |- (A.x e. A E!yph -> A.xA.x e. A E!yph)
2		ra4 1670	. . . . . . . 8 |- (A.x e. A E!yph -> (x e. A -> E!yph))
3		eumo 1388	. . . . . . . . . 10 |- (E!yph -> E*yph)
4	3	imim2i 17	. . . . . . . . 9 |- ((x e. A -> E!yph) -> (x e. A -> E*yph))
5		moanimv 1406	. . . . . . . . 9 |- (E*y(x e. A /\ ph) <-> (x e. A -> E*yph))
6	4, 5	sylibr 200	. . . . . . . 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 974	. . . . . 6 |- (A.x e. A E!yph -> A.xE*y(x e. A /\ ph))
9		funopab 3488	. . . . . 6 |- (Fun {<.x, y>. | (x e. A /\ ph)} <-> A.xE*y(x e. A /\ ph))
10	8, 9	sylibr 200	. . . . 5 |- (A.x e. A E!yph -> Fun {<.x, y>. | (x e. A /\ ph)})
11		euex 1371	. . . . . . 7 |- (E!yph -> E.yph)
12	11	r19.20si 1682	. . . . . 6 |- (A.x e. A E!yph -> A.x e. A E.yph)
13		dmopab3 3279	. . . . . 6 |- (A.x e. A E.yph <-> dom {<.x, y>. | (x e. A /\ ph)} = A)
14	12, 13	sylib 198	. . . . 5 |- (A.x e. A E!yph -> dom {<.x, y>. | (x e. A /\ ph)} = A)
15	10, 14	jca 288	. . . 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 3156	. . . 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 200	. . 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 3522	. . . 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 200	. 2 |- (A.x e. A E!yph -> F Fn A)
22		hbopab1 2775	. . . . 5 |- (z e. {<.x, y>. | (x e. A /\ ph)} -> A.x z e. {<.x, y>. | (x e. A /\ ph)})
23	18, 22	hbxfr 1539	. . . 4 |- (z e. F -> A.x z e. F)
24		ax-17 1190	. . . 4 |- (z e. A -> A.x z e. A)
25	23, 24	hbfn 3524	. . 3 |- (F Fn A -> A.x F Fn A)
26		fneu2 3533	. . . . . 6 |- ((F Fn A /\ x e. A) -> E!z<.x, z>. e. F)
27		ax-17 1190	. . . . . . . 8 |- (w e. <.x, z>. -> A.y w e. <.x, z>.)
28		hbopab2 2776	. . . . . . . . 9 |- (z e. {<.x, y>. | (x e. A /\ ph)} -> A.y z e. {<.x, y>. | (x e. A /\ ph)})
29	18, 28	hbxfr 1539	. . . . . . . 8 |- (z e. F -> A.y z e. F)
30	27, 29	hbel 1542	. . . . . . 7 |- (<.x, z>. e. F -> A.y<.x, z>. e. F)
31		ax-17 1190	. . . . . . 7 |- (<.x, y>. e. F -> A.z<.x, y>. e. F)
32		opeq2 2457	. . . . . . . 8 |- (z = y -> <.x, z>. = <.x, y>.)
33	32	eleq1d 1516	. . . . . . 7 |- (z = y -> (<.x, z>. e. F <-> <.x, y>. e. F))
34	30, 31, 33	cbveu 1368	. . . . . 6 |- (E!z<.x, z>. e. F <-> E!y<.x, y>. e. F)
35	26, 34	sylib 198	. . . . 5 |- ((F Fn A /\ x e. A) -> E!y<.x, y>. e. F)
36	18	eleq2i 1514	. . . . . . . . 9 |- (<.x, y>. e. F <-> <.x, y>. e. {<.x, y>. | (x e. A /\ ph)})
37		opabid 2772	. . . . . . . . 9 |- (<.x, y>. e. {<.x, y>. | (x e. A /\ ph)} <-> (x e. A /\ ph))
38	36, 37	bitr 173	. . . . . . . 8 |- (<.x, y>. e. F <-> (x e. A /\ ph))
39	38	eubii 1364	. . . . . . 7 |- (E!y<.x, y>. e. F <-> E!y(x e. A /\ ph))
40		euanv 1409	. . . . . . 7 |- (E!y(x e. A /\ ph) <-> (x e. A /\ E!yph))
41	39, 40	bitr 173	. . . . . 6 |- (E!y<.x, y>. e. F <-> (x e. A /\ E!yph))
42	41	pm3.27bi 326	. . . . 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 373	. . 3 |- (F Fn A -> (x e. A -> E!yph))
45	25, 44	r19.21ai 1688	. 2 |- (F Fn A -> A.x e. A E!yph)
46	21, 45	impbi 157	1 |- (A.x e. A E!yph <-> F Fn A)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 950  E.wex 956   = wceq 1099   e. wcel 1105  E!weu 1357  E*wmo 1358  A.wral 1621  <.cop 2382  {copab 2634  dom cdm 3133  Fun wfun 3139   Fn wfn 3140

This theorem is referenced by:  fnopab2g 3556  fnopab 3557  elrnopabg 3739  fopab2 3762  en2d 4335

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-13 1107  ax-14 1108  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-sep 2671  ax-pow 2710  ax-pr 2747

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-ral 1625  df-v 1787  df-dif 2020  df-un 2021  df-in 2022  df-ss 2024  df-nul 2252  df-pw 2373  df-sn 2383  df-pr 2384  df-op 2387  df-br 2588  df-opab 2635  df-id 2797  df-xp 3147  df-rel 3148  df-cnv 3149  df-co 3150  df-dm 3151  df-rn 3152  df-fun 3155  df-fn 3156

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