Theorem fopab2 3808

Description: Functionality of an ordered-pair class abstraction.

Hypothesis

Ref	Expression
fopab2.1	|- F = {<.x, y>. | (x e. A /\ y = C)}

Assertion

Ref	Expression
fopab2	|- (A.x e. A C e. B <-> F:A-->B)

Distinct variable groups:   x,y,A   x,B,y   y,C

Proof of Theorem fopab2

Step	Hyp	Ref	Expression
1		elisset 1808	. . . . . . 7 |- (C e. B -> C e. V)
2		eueq 1907	. . . . . . 7 |- (C e. V <-> E!y y = C)
3	1, 2	sylib 198	. . . . . 6 |- (C e. B -> E!y y = C)
4	3	r19.20si 1698	. . . . 5 |- (A.x e. A C e. B -> A.x e. A E!y y = C)
5		fopab2.1	. . . . . 6 |- F = {<.x, y>. | (x e. A /\ y = C)}
6	5	fnopabg 3601	. . . . 5 |- (A.x e. A E!y y = C <-> F Fn A)
7	4, 6	sylib 198	. . . 4 |- (A.x e. A C e. B -> F Fn A)
8		hbra1 1679	. . . . . . . . 9 |- (A.x e. A C e. B -> A.xA.x e. A C e. B)
9		ax-17 968	. . . . . . . . 9 |- (y e. B -> A.x y e. B)
10		ra4 1686	. . . . . . . . . 10 |- (A.x e. A C e. B -> (x e. A -> C e. B))
11		eleq1a 1535	. . . . . . . . . . . 12 |- (C e. B -> (y = C -> y e. B))
12	11	imim2i 17	. . . . . . . . . . 11 |- ((x e. A -> C e. B) -> (x e. A -> (y = C -> y e. B)))
13	12	imp3a 361	. . . . . . . . . 10 |- ((x e. A -> C e. B) -> ((x e. A /\ y = C) -> y e. B))
14	10, 13	syl 10	. . . . . . . . 9 |- (A.x e. A C e. B -> ((x e. A /\ y = C) -> y e. B))
15	8, 9, 14	19.23ad 1062	. . . . . . . 8 |- (A.x e. A C e. B -> (E.x(x e. A /\ y = C) -> y e. B))
16		rnopab 3339	. . . . . . . . 9 |- ran {<.x, y>. | (x e. A /\ y = C)} = {y | E.x(x e. A /\ y = C)}
17	16	abeq2i 1562	. . . . . . . 8 |- (y e. ran {<.x, y>. | (x e. A /\ y = C)} <-> E.x(x e. A /\ y = C))
18	15, 17	syl5ib 206	. . . . . . 7 |- (A.x e. A C e. B -> (y e. ran {<.x, y>. | (x e. A /\ y = C)} -> y e. B))
19	18	19.21aiv 1281	. . . . . 6 |- (A.x e. A C e. B -> A.y(y e. ran {<.x, y>. | (x e. A /\ y = C)} -> y e. B))
20		hbopab2 2803	. . . . . . . 8 |- (z e. {<.x, y>. | (x e. A /\ y = C)} -> A.y z e. {<.x, y>. | (x e. A /\ y = C)})
21	20	hbrn 3337	. . . . . . 7 |- (z e. ran {<.x, y>. | (x e. A /\ y = C)} -> A.y z e. ran {<.x, y>. | (x e. A /\ y = C)})
22		ax-17 968	. . . . . . 7 |- (z e. B -> A.y z e. B)
23	21, 22	dfss2f 2050	. . . . . 6 |- (ran {<.x, y>. | (x e. A /\ y = C)} (_ B <-> A.y(y e. ran {<.x, y>. | (x e. A /\ y = C)} -> y e. B))
24	19, 23	sylibr 200	. . . . 5 |- (A.x e. A C e. B -> ran {<.x, y>. | (x e. A /\ y = C)} (_ B)
25	5	rneqi 3329	. . . . 5 |- ran F = ran {<.x, y>. | (x e. A /\ y = C)}
26	24, 25	syl5ss 2095	. . . 4 |- (A.x e. A C e. B -> ran F (_ B)
27	7, 26	jca 288	. . 3 |- (A.x e. A C e. B -> (F Fn A /\ ran F (_ B))
28		df-f 3184	. . 3 |- (F:A-->B <-> (F Fn A /\ ran F (_ B))
29	27, 28	sylibr 200	. 2 |- (A.x e. A C e. B -> F:A-->B)
30		fdm 3617	. . . 4 |- (F:A-->B -> dom F = A)
31		dmopab3 3311	. . . . 5 |- (A.x e. A E.y y = C <-> dom {<.x, y>. | (x e. A /\ y = C)} = A)
32		isset 1805	. . . . . 6 |- (C e. V <-> E.y y = C)
33	32	ralbii 1659	. . . . 5 |- (A.x e. A C e. V <-> A.x e. A E.y y = C)
34	5	dmeqi 3301	. . . . . 6 |- dom F = dom {<.x, y>. | (x e. A /\ y = C)}
35	34	eqeq1i 1474	. . . . 5 |- (dom F = A <-> dom {<.x, y>. | (x e. A /\ y = C)} = A)
36	31, 33, 35	3bitr4r 184	. . . 4 |- (dom F = A <-> A.x e. A C e. V)
37	30, 36	sylib 198	. . 3 |- (F:A-->B -> A.x e. A C e. V)
38		hbopab1 2802	. . . . . 6 |- (z e. {<.x, y>. | (x e. A /\ y = C)} -> A.x z e. {<.x, y>. | (x e. A /\ y = C)})
39		ax-17 968	. . . . . 6 |- (z e. A -> A.x z e. A)
40		ax-17 968	. . . . . 6 |- (z e. B -> A.x z e. B)
41	38, 39, 40	hbf 3611	. . . . 5 |- ({<.x, y>. | (x e. A /\ y = C)}:A-->B -> A.x{<.x, y>. | (x e. A /\ y = C)}:A-->B)
42		feq1 3606	. . . . . 6 |- (F = {<.x, y>. | (x e. A /\ y = C)} -> (F:A-->B <-> {<.x, y>. | (x e. A /\ y = C)}:A-->B))
43	5, 42	ax-mp 7	. . . . 5 |- (F:A-->B <-> {<.x, y>. | (x e. A /\ y = C)}:A-->B)
44	43	albii 996	. . . . 5 |- (A.x F:A-->B <-> A.x{<.x, y>. | (x e. A /\ y = C)}:A-->B)
45	41, 43, 44	3imtr4 219	. . . 4 |- (F:A-->B -> A.x F:A-->B)
46		ffvelrn 3799	. . . . . . 7 |- ((F:A-->B /\ x e. A) -> (F` x) e. B)
47	46	adantr 389	. . . . . 6 |- (((F:A-->B /\ x e. A) /\ C e. V) -> (F` x) e. B)
48		fvopab2 3776	. . . . . . . . 9 |- ((x e. A /\ C e. V) -> ({<.x, y>. | (x e. A /\ y = C)}` x) = C)
49	5	fveq1i 3710	. . . . . . . . 9 |- (F` x) = ({<.x, y>. | (x e. A /\ y = C)}` x)
50	48, 49	syl5eq 1511	. . . . . . . 8 |- ((x e. A /\ C e. V) -> (F` x) = C)
51	50	eleq1d 1532	. . . . . . 7 |- ((x e. A /\ C e. V) -> ((F` x) e. B <-> C e. B))
52	51	adantll 392	. . . . . 6 |- (((F:A-->B /\ x e. A) /\ C e. V) -> ((F` x) e. B <-> C e. B))
53	47, 52	mpbid 195	. . . . 5 |- (((F:A-->B /\ x e. A) /\ C e. V) -> C e. B)
54	53	ex 373	. . . 4 |- ((F:A-->B /\ x e. A) -> (C e. V -> C e. B))
55	45, 54	r19.20da 1700	. . 3 |- (F:A-->B -> (A.x e. A C e. V -> A.x e. A C e. B))
56	37, 55	mpd 26	. 2 |- (F:A-->B -> A.x e. A C e. B)
57	29, 56	impbi 157	1 |- (A.x e. A C e. B <-> F:A-->B)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 951   = wceq 953   e. wcel 955  E.wex 977  E!weu 1373  A.wral 1637  Vcvv 1802   (_ wss 2037  {copab 2656  dom cdm 3160  ran crn 3161   Fn wfn 3167  -->wf 3168  ` cfv 3172

This theorem is referenced by:  fopabssxp 3809  rnssopab 3810  fopab3 3811  fopab 3812  f1stres 4077  f2ndres 4078  curry1f 4083  foprab2 4103  dom2d 4385  mapenlem2 4470  xpmapenlem4 4479  ser1cl2 6270  cvgratlem5 7189  negfcncf 7204  mulc1cncf 7214  efseq0ex 7253  lmfexlem1 7891  metcnp4 7904  xplmi 7907  xpcn 7910  bopcnlem4 7918  grplactf1o 8034  sqcn 8270  va1cnlem 8279  ipblnfi 8447  ubthlem3 8462  sincolem 8584  occllem4 9092  projlem24 9125  hoaddclt 9601  homulclt 9602  brafnt 9787  kbopt 9793  cnlnadjlem2 9916  strlem3a 10089  hstrlem3a 10097  cayleylem2 10317  fopab2a 10362

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-fv 3188

Copyright terms: Public domain