Theorem rnssopab 3825

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 (_ 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 3823	. . 3 |- (A.x e. A C e. B <-> F:A-->B)
3		frn 3633	. . 3 |- (F:A-->B -> ran F (_ B)
4	2, 3	sylbi 199	. 2 |- (A.x e. A C e. B -> ran F (_ B)
5		hbopab1 2813	. . . . . 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 1563	. . . . 5 |- (z e. F -> A.x z e. F)
7	6	hbrn 3351	. . . 4 |- (z e. ran F -> A.x z e. ran F)
8		ax-17 971	. . . 4 |- (z e. B -> A.x z e. B)
9	7, 8	hbss 2062	. . 3 |- (ran F (_ B -> A.xran F (_ B)
10		ssel 2063	. . . 4 |- (ran F (_ B -> (C e. ran F -> C e. B))
11		rnssopab.2	. . . . . . 7 |- C e. V
12		fvopab2 3791	. . . . . . 7 |- ((x e. A /\ C e. V) -> ({<.x, y>. | (x e. A /\ y = C)}` x) = C)
13	11, 12	mpan2 696	. . . . . 6 |- (x e. A -> ({<.x, y>. | (x e. A /\ y = C)}` x) = C)
14	1	fveq1i 3725	. . . . . 6 |- (F` x) = ({<.x, y>. | (x e. A /\ y = C)}` x)
15	13, 14	syl5eq 1519	. . . . 5 |- (x e. A -> (F` x) = C)
16	11, 1	fnopab2 3618	. . . . . 6 |- F Fn A
17		fnfvelrn 3813	. . . . . 6 |- ((F Fn A /\ x e. A) -> (F` x) e. ran F)
18	16, 17	mpan 695	. . . . 5 |- (x e. A -> (F` x) e. ran F)
19	15, 18	eqeltrrd 1549	. . . 4 |- (x e. A -> C e. ran F)
20	10, 19	syl5 21	. . 3 |- (ran F (_ B -> (x e. A -> C e. B))
21	9, 20	r19.21ai 1712	. 2 |- (ran F (_ B -> A.x e. A C e. B)
22	4, 21	impbi 157	1 |- (A.x e. A C e. B <-> ran F (_ B)

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645  Vcvv 1811   (_ wss 2047  {copab 2666  ran crn 3171   Fn wfn 3177  -->wf 3178  ` cfv 3182

This theorem is referenced by:  fopab3 3826  oprcn 7977  ip1cnilem2 8374  ip1cnilem3 8375  ipasslem6 8495  kbass2t 10050

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198

Copyright terms: Public domain