Theorem fnopabg 2745

Description: Functionality and domain of an ordered pair 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(s):   x,y,A

Proof of Theorem fnopabg

Step	Hyp	Ref	Expression
1		hbra1 1237	. . . . . . 7 |- (A.x e. A E!yph -> A.xA.x e. A E!yph)
2		ra4 1243	. . . . . . . 8 |- (A.x e. A E!yph -> (x e. A -> E!yph))
3		eumo 1037	. . . . . . . . . 10 |- (E!yph -> E*yph)
4	3	syl3 18	. . . . . . . . 9 |- ((x e. A -> E!yph) -> (x e. A -> E*yph))
5		moanimv 1052	. . . . . . . . 9 |- (E*y(x e. A /\ ph) <-> (x e. A -> E*yph))
6	4, 5	sylibr 175	. . . . . . . 8 |- ((x e. A -> E!yph) -> E*y(x e. A /\ ph))
7	2, 6	syl 12	. . . . . . 7 |- (A.x e. A E!yph -> E*y(x e. A /\ ph))
8	1, 7	19.21ai 740	. . . . . 6 |- (A.x e. A E!yph -> A.xE*y(x e. A /\ ph))
9		funopab 2694	. . . . . 6 |- (Fun {<.x, y>. | (x e. A /\ ph)} <-> A.xE*y(x e. A /\ ph))
10	8, 9	sylibr 175	. . . . 5 |- (A.x e. A E!yph -> Fun {<.x, y>. | (x e. A /\ ph)})
11		euex 1021	. . . . . . 7 |- (E!yph -> E.yph)
12	11	r19.20si 1254	. . . . . 6 |- (A.x e. A E!yph -> A.x e. A E.yph)
13		dmopab2 2541	. . . . . 6 |- (A.x e. A E.yph <-> dom {<.x, y>. | (x e. A /\ ph)} = A)
14	12, 13	sylib 173	. . . . 5 |- (A.x e. A E!yph -> dom {<.x, y>. | (x e. A /\ ph)} = A)
15	10, 14	jca 236	. . . 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 2433	. . . 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 175	. . 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 2718	. . . 4 |- (F = {<.x, y>. | (x e. A /\ ph)} -> (F Fn A <-> {<.x, y>. | (x e. A /\ ph)} Fn A))
20	18, 19	ax-mp 6	. . 3 |- (F Fn A <-> {<.x, y>. | (x e. A /\ ph)} Fn A)
21	17, 20	sylibr 175	. 2 |- (A.x e. A E!yph -> F Fn A)
22		hbopab1 2112	. . . . 5 |- (z e. {<.x, y>. | (x e. A /\ ph)} -> A.x z e. {<.x, y>. | (x e. A /\ ph)})
23	18	eleq2i 1153	. . . . 5 |- (z e. F <-> z e. {<.x, y>. | (x e. A /\ ph)})
24	23	bial 695	. . . . 5 |- (A.x z e. F <-> A.x z e. {<.x, y>. | (x e. A /\ ph)})
25	22, 23, 24	3imtr4 192	. . . 4 |- (z e. F -> A.x z e. F)
26		ax-17 925	. . . 4 |- (z e. A -> A.x z e. A)
27	25, 26	hbfn 2720	. . 3 |- (F Fn A -> A.x F Fn A)
28		fneu2 2729	. . . . . 6 |- ((F Fn A /\ x e. A) -> E!z<.x, z>. e. F)
29		ax-17 925	. . . . . . . 8 |- (w e. <.x, z>. -> A.y w e. <.x, z>.)
30		hbopab2 2113	. . . . . . . . 9 |- (z e. {<.x, y>. | (x e. A /\ ph)} -> A.y z e. {<.x, y>. | (x e. A /\ ph)})
31	23	bial 695	. . . . . . . . 9 |- (A.y z e. F <-> A.y z e. {<.x, y>. | (x e. A /\ ph)})
32	30, 23, 31	3imtr4 192	. . . . . . . 8 |- (z e. F -> A.y z e. F)
33	29, 32	hbel 1172	. . . . . . 7 |- (<.x, z>. e. F -> A.y<.x, z>. e. F)
34		ax-17 925	. . . . . . 7 |- (<.x, y>. e. F -> A.z<.x, y>. e. F)
35		opeq2 1877	. . . . . . . 8 |- (z = y -> <.x, z>. = <.x, y>.)
36	35	eleq1d 1155	. . . . . . 7 |- (z = y -> (<.x, z>. e. F <-> <.x, y>. e. F))
37	33, 34, 36	cbveu 1018	. . . . . 6 |- (E!z<.x, z>. e. F <-> E!y<.x, y>. e. F)
38	28, 37	sylib 173	. . . . 5 |- ((F Fn A /\ x e. A) -> E!y<.x, y>. e. F)
39	18	eleq2i 1153	. . . . . . . . 9 |- (<.x, y>. e. F <-> <.x, y>. e. {<.x, y>. | (x e. A /\ ph)})
40		opabid 2099	. . . . . . . . 9 |- (<.x, y>. e. {<.x, y>. | (x e. A /\ ph)} <-> (x e. A /\ ph))
41	39, 40	bitr 151	. . . . . . . 8 |- (<.x, y>. e. F <-> (x e. A /\ ph))
42	41	bieu 1014	. . . . . . 7 |- (E!y<.x, y>. e. F <-> E!y(x e. A /\ ph))
43		euanv 1053	. . . . . . 7 |- (E!y(x e. A /\ ph) <-> (x e. A /\ E!yph))
44	42, 43	bitr 151	. . . . . 6 |- (E!y<.x, y>. e. F <-> (x e. A /\ E!yph))
45	44	pm3.27bd 263	. . . . 5 |- (E!y<.x, y>. e. F -> E!yph)
46	38, 45	syl 12	. . . 4 |- ((F Fn A /\ x e. A) -> E!yph)
47	46	exp 291	. . 3 |- (F Fn A -> (x e. A -> E!yph))
48	27, 47	r19.21ai 1258	. 2 |- (F Fn A -> A.x e. A E!yph)
49	21, 48	impbi 139	1 |- (A.x e. A E!yph <-> F Fn A)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678   = weq 797  E!weu 1007  E*wmo 1008   = wceq 1091   e. wcel 1092  A.wral 1201  <.cop 1810  {copab 2055  dom cdm 2410  Fun wfun 2416   Fn wfn 2417

This theorem is referenced by:  fnopab 2746  fopab2 2891  en2d 3303

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-fun 2432  df-fn 2433

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