Theorem fopab2 3818

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 1814	. . . . . . 7 |- (C e. B -> C e. V)
2		eueq 1913	. . . . . . 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 1704	. . . . 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 3611	. . . . 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 1685	. . . . . . . . 9 |- (A.x e. A C e. B -> A.xA.x e. A C e. B)
9		ax-17 970	. . . . . . . . 9 |- (y e. B -> A.x y e. B)
10		ra4 1692	. . . . . . . . . 10 |- (A.x e. A C e. B -> (x e. A -> C e. B))
11		eleq1a 1541	. . . . . . . . . . . 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 1065	. . . . . . . 8 |- (A.x e. A C e. B -> (E.x(x e. A /\ y = C) -> y e. B))
16		rnopab 3349	. . . . . . . . 9 |- ran {<.x, y>. | (x e. A /\ y = C)} = {y | E.x(x e. A /\ y = C)}
17	16	abeq2i 1568	. . . . . . . 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 1285	. . . . . 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 2810	. . . . . . . 8 |- (z e. {<.x, y>. | (x e. A /\ y = C)} -> A.y z e. {<.x, y>. | (x e. A /\ y = C)})
21	20	hbrn 3347	. . . . . . 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-17 970	. . . . . . 7 |- (z e. B -> A.y z e. B)
23	21, 22	dfss2f 2057	. . . . . 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 3336	. . . . 5 |- ran F = ran {<.x, y>. | (x e. A /\ y = C)}
26	24, 25	syl5ss 2102	. . . 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 3190	. . 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 3627	. . . 4 |- (F:A-->B -> dom F = A)
31		dmopab3 3318	. . . . 5 |- (A.x e. A E.y y = C <-> dom {<.x, y>. | (x e. A /\ y = C)} = A)
32		isset 1811	. . . . . 6 |- (C e. V <-> E.y y = C)
33	32	ralbii 1665	. . . . 5 |- (A.x e. A C e. V <-> A.x e. A E.y y = C)
34	5	dmeqi 3308	. . . . . 6 |- dom F = dom {<.x, y>. | (x e. A /\ y = C)}
35	34	eqeq1i 1480	. . . . 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 2809	. . . . . 6 |- (z e. {<.x, y>. | (x e. A /\ y = C)} -> A.x z e. {<.x, y>. | (x e. A /\ y = C)})
39		ax-17 970	. . . . . 6 |- (z e. A -> A.x z e. A)
40		ax-17 970	. . . . . 6 |- (z e. B -> A.x z e. B)
41	38, 39, 40	hbf 3621	. . . . 5 |- ({<.x, y>. | (x e. A /\ y = C)}:A-->B -> A.x{<.x, y>. | (x e. A /\ y = C)}:A-->B)
42		feq1 3616	. . . . . 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 998	. . . . 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 3809	. . . . . . 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 3786	. . . . . . . . 9 |- ((x e. A /\ C e. V) -> ({<.x, y>. | (x e. A /\ y = C)}` x) = C)
49	5	fveq1i 3720	. . . . . . . . 9 |- (F` x) = ({<.x, y>. | (x e. A /\ y = C)}` x)
50	48, 49	syl5eq 1517	. . . . . . . 8 |- ((x e. A /\ C e. V) -> (F` x) = C)
51	50	eleq1d 1538	. . . . . . 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 1706	. . 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 953   = wceq 955   e. wcel 957  E.wex 979  E!weu 1379  A.wral 1643  Vcvv 1808   (_ wss 2044  {copab 2662  dom cdm 3166  ran crn 3167   Fn wfn 3173  -->wf 3174  ` cfv 3178

This theorem is referenced by:  fopabssxp 3819  rnssopab 3820  fopab3 3821  fopab 3822  f1stres 4086  f2ndres 4087  curry1f 4092  foprab2 4112  dom2d 4394  mapenlem2 4479  xpmapenlem4 4488  ser1cl2 6283  cvgratlem5 7206  negfcncf 7221  mulc1cncf 7231  efseq0ex 7270  lmfexlem1 7918  metcnp4 7932  xplmi 7935  xpcn 7938  bopcnlem4 7946  grplactf1o 8061  sqcn 8298  va1cnlem 8307  ipblnfi 8475  ubthlem3 8490  sincolem 8619  occllem4 9131  projlem24 9164  hoaddclt 9641  homulclt 9642  brafnt 9828  kbopt 9834  cnlnadjlem2 9957  strlem3a 10135  hstrlem3a 10143  cayleylem2 10366  fopab2a 10420

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-fv 3194

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