Theorem fopab2 2891

Description: Functionality of an ordered pair 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 group(s):   x,y,A   x,B,y   y,C

Proof of Theorem fopab2

Step	Hyp	Ref	Expression
1		elisset 1354	. . . . . . 7 |- (C e. B -> C e. V)
2		eueq 1427	. . . . . . 7 |- (C e. V <-> E!y y = C)
3	1, 2	sylib 173	. . . . . 6 |- (C e. B -> E!y y = C)
4	3	r19.20si 1254	. . . . 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 2745	. . . . 5 |- (A.x e. A E!y y = C <-> F Fn A)
7	4, 6	sylib 173	. . . 4 |- (A.x e. A C e. B -> F Fn A)
8		hbra1 1237	. . . . . . . . 9 |- (A.x e. A C e. B -> A.xA.x e. A C e. B)
9		ax-17 925	. . . . . . . . 9 |- (y e. B -> A.x y e. B)
10		ra4 1243	. . . . . . . . . 10 |- (A.x e. A C e. B -> (x e. A -> C e. B))
11		eleq1a 1158	. . . . . . . . . . . 12 |- (C e. B -> (y = C -> y e. B))
12	11	syl3 18	. . . . . . . . . . 11 |- ((x e. A -> C e. B) -> (x e. A -> (y = C -> y e. B)))
13	12	imp3a 279	. . . . . . . . . 10 |- ((x e. A -> C e. B) -> ((x e. A /\ y = C) -> y e. B))
14	10, 13	syl 12	. . . . . . . . 9 |- (A.x e. A C e. B -> ((x e. A /\ y = C) -> y e. B))
15	8, 9, 14	19.23ad 748	. . . . . . . 8 |- (A.x e. A C e. B -> (E.x(x e. A /\ y = C) -> y e. B))
16		rnopab 2566	. . . . . . . . 9 |- ran {<.x, y>. | (x e. A /\ y = C)} = {y | E.x(x e. A /\ y = C)}
17	16	cleqabi 1176	. . . . . . . 8 |- (y e. ran {<.x, y>. | (x e. A /\ y = C)} <-> E.x(x e. A /\ y = C))
18	15, 17	syl5ib 181	. . . . . . 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 943	. . . . . 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 2113	. . . . . . . 8 |- (z e. {<.x, y>. | (x e. A /\ y = C)} -> A.y z e. {<.x, y>. | (x e. A /\ y = C)})
21	20	hbrn 2564	. . . . . . 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-17 925	. . . . . . 7 |- (z e. B -> A.y z e. B)
23	21, 22	dfss2f 1499	. . . . . 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 175	. . . . 5 |- (A.x e. A C e. B -> ran {<.x, y>. | (x e. A /\ y = C)} (_ B)
25	5	rneqi 2556	. . . . 5 |- ran F = ran {<.x, y>. | (x e. A /\ y = C)}
26	24, 25	syl5ss 1544	. . . 4 |- (A.x e. A C e. B -> ran F (_ B)
27	7, 26	jca 236	. . 3 |- (A.x e. A C e. B -> (F Fn A /\ ran F (_ B))
28		df-f 2434	. . 3 |- (F:A-->B <-> (F Fn A /\ ran F (_ B))
29	27, 28	sylibr 175	. 2 |- (A.x e. A C e. B -> F:A-->B)
30		fdm 2756	. . . 4 |- (F:A-->B -> dom F = A)
31		dmopab2 2541	. . . . 5 |- (A.x e. A E.y y = C <-> dom {<.x, y>. | (x e. A /\ y = C)} = A)
32		isset 1351	. . . . . 6 |- (C e. V <-> E.y y = C)
33	32	biral 1223	. . . . 5 |- (A.x e. A C e. V <-> A.x e. A E.y y = C)
34	5	dmeqi 2532	. . . . . 6 |- dom F = dom {<.x, y>. | (x e. A /\ y = C)}
35	34	cleq1i 1108	. . . . 5 |- (dom F = A <-> dom {<.x, y>. | (x e. A /\ y = C)} = A)
36	31, 33, 35	3bitr4r 159	. . . 4 |- (dom F = A <-> A.x e. A C e. V)
37	30, 36	sylib 173	. . 3 |- (F:A-->B -> A.x e. A C e. V)
38		hbopab1 2112	. . . . . 6 |- (z e. {<.x, y>. | (x e. A /\ y = C)} -> A.x z e. {<.x, y>. | (x e. A /\ y = C)})
39		ax-17 925	. . . . . 6 |- (z e. A -> A.x z e. A)
40		ax-17 925	. . . . . 6 |- (z e. B -> A.x z e. B)
41	38, 39, 40	hbf 2751	. . . . 5 |- ({<.x, y>. | (x e. A /\ y = C)}:A-->B -> A.x{<.x, y>. | (x e. A /\ y = C)}:A-->B)
42		feq1 2748	. . . . . 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 6	. . . . 5 |- (F:A-->B <-> {<.x, y>. | (x e. A /\ y = C)}:A-->B)
44	43	bial 695	. . . . 5 |- (A.x F:A-->B <-> A.x{<.x, y>. | (x e. A /\ y = C)}:A-->B)
45	41, 43, 44	3imtr4 192	. . . 4 |- (F:A-->B -> A.x F:A-->B)
46		ffvrn 2890	. . . . . . 7 |- ((F:A-->B /\ x e. A) -> (F` x) e. B)
47	46	adantr 306	. . . . . 6 |- (((F:A-->B /\ x e. A) /\ C e. V) -> (F` x) e. B)
48		fvopab2 2878	. . . . . . . . 9 |- ((x e. A /\ C e. V) -> ({<.x, y>. | (x e. A /\ y = C)}` x) = C)
49	5	fveq1i 2833	. . . . . . . . 9 |- (F` x) = ({<.x, y>. | (x e. A /\ y = C)}` x)
50	48, 49	syl5eq 1136	. . . . . . . 8 |- ((x e. A /\ C e. V) -> (F` x) = C)
51	50	eleq1d 1155	. . . . . . 7 |- ((x e. A /\ C e. V) -> ((F` x) e. B <-> C e. B))
52	51	adantll 309	. . . . . 6 |- (((F:A-->B /\ x e. A) /\ C e. V) -> ((F` x) e. B <-> C e. B))
53	47, 52	mpbid 170	. . . . 5 |- (((F:A-->B /\ x e. A) /\ C e. V) -> C e. B)
54	53	exp 291	. . . 4 |- ((F:A-->B /\ x e. A) -> (C e. V -> C e. B))
55	45, 54	r19.20da 1255	. . 3 |- (F:A-->B -> (A.x e. A C e. V -> A.x e. A C e. B))
56	37, 55	mpd 46	. 2 |- (F:A-->B -> A.x e. A C e. B)
57	29, 56	impbi 139	1 |- (A.x e. A C e. B <-> F:A-->B)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678  E!weu 1007   = wceq 1091   e. wcel 1092  A.wral 1201  Vcvv 1348   (_ wss 1487  {copab 2055  dom cdm 2410  ran crn 2411   Fn wfn 2417  -->wf 2418  ` cfv 2422

This theorem is referenced by:  f1stres 3096  dom2d 3307  pw2en 3348  mapenlem2 3385  xpmapenlem4 3394  occllem4 5183  projlem24 5216  hosf 5602  hodf 5603  strlem3a 5693

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-un 1076  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-rex 1206  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-uni 1920  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-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-fv 2438

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