Theorem rnssopab 4798

Description: Range of a function that is expressed as an ordered-pair class abstraction.

Hypotheses

Ref	Expression
fopab2.1	|- F = {<.x, y>. | (x e. A /\ y = C)}
rnssopab.2	|- C e. _V

Assertion

Ref	Expression
rnssopab	|- (A.x e. A C e. B <-> ran F C_ B)

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

Proof of Theorem rnssopab

Step	Hyp	Ref	Expression
1		fopab2.1	. . . 4 |- F = {<.x, y>. | (x e. A /\ y = C)}
2	1	fopab2 4796	. . 3 |- (A.x e. A C e. B <-> F:A-->B)
3		frn 4569	. . 3 |- (F:A-->B -> ran F C_ B)
4	2, 3	sylbi 216	. 2 |- (A.x e. A C e. B -> ran F C_ B)
5		hbopab1 3562	. . . . . 6 |- (z e. {<.x, y>. | (x e. A /\ y = C)} -> A.x z e. {<.x, y>. | (x e. A /\ y = C)})
6	1, 5	hbxfr 1992	. . . . 5 |- (z e. F -> A.x z e. F)
7	6	hbrn 4198	. . . 4 |- (z e. ran F -> A.x z e. ran F)
8		ax-17 1317	. . . 4 |- (z e. B -> A.x z e. B)
9	7, 8	hbss 2614	. . 3 |- (ran F C_ B -> A.xran F C_ B)
10		ssel 2615	. . . 4 |- (ran F C_ B -> (C e. ran F -> C e. B))
11		rnssopab.2	. . . . . . 7 |- C e. _V
12		fvopab2 4754	. . . . . . 7 |- ((x e. A /\ C e. _V) -> ({<.x, y>. | (x e. A /\ y = C)}` x) = C)
13	11, 12	mpan2 760	. . . . . 6 |- (x e. A -> ({<.x, y>. | (x e. A /\ y = C)}` x) = C)
14	1	fveq1i 4682	. . . . . 6 |- (F` x) = ({<.x, y>. | (x e. A /\ y = C)}` x)
15	13, 14	syl5eq 1940	. . . . 5 |- (x e. A -> (F` x) = C)
16	11, 1	fnopab2 4549	. . . . . 6 |- F Fn A
17		fnfvelrn 4786	. . . . . 6 |- ((F Fn A /\ x e. A) -> (F` x) e. ran F)
18	16, 17	mpan 759	. . . . 5 |- (x e. A -> (F` x) e. ran F)
19	15, 18	eqeltrrd 1972	. . . 4 |- (x e. A -> C e. ran F)
20	10, 19	syl5 20	. . 3 |- (ran F C_ B -> (x e. A -> C e. B))
21	9, 20	r19.21ai 2174	. 2 |- (ran F C_ B -> A.x e. A C e. B)
22	4, 21	impbii 174	1 |- (A.x e. A C e. B <-> ran F C_ B)

Syntax hints:   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292   C_ wss 2593  {copab 3395  ran crn 3987   Fn wfn 3993  -->wf 3994  ` cfv 3998

This theorem is referenced by:  fopab3 4799  oprcn 9255  ip1cnilem2 9713  ip1cnilem3 9714  ipasslem6 9836  kbass2 11688  supbrr 15048  pcocn 16076

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014

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