Theorem rnssopab 3816

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 3814	. . 3 |- (A.x e. A C e. B <-> F:A-->B)
3		frn 3624	. . 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 2808	. . . . . 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 1560	. . . . 5 |- (z e. F -> A.x z e. F)
7	6	hbrn 3345	. . . 4 |- (z e. ran F -> A.x z e. ran F)
8		ax-17 969	. . . 4 |- (z e. B -> A.x z e. B)
9	7, 8	hbss 2058	. . 3 |- (ran F (_ B -> A.xran F (_ B)
10		ssel 2059	. . . 4 |- (ran F (_ B -> (C e. ran F -> C e. B))
11		rnssopab.2	. . . . . . 7 |- C e. V
12		fvopab2 3782	. . . . . . 7 |- ((x e. A /\ C e. V) -> ({<.x, y>. | (x e. A /\ y = C)}` x) = C)
13	11, 12	mpan2 695	. . . . . 6 |- (x e. A -> ({<.x, y>. | (x e. A /\ y = C)}` x) = C)
14	1	fveq1i 3716	. . . . . 6 |- (F` x) = ({<.x, y>. | (x e. A /\ y = C)}` x)
15	13, 14	syl5eq 1516	. . . . 5 |- (x e. A -> (F` x) = C)
16	11, 1	fnopab2 3610	. . . . . 6 |- F Fn A
17		fnfvelrn 3804	. . . . . 6 |- ((F Fn A /\ x e. A) -> (F` x) e. ran F)
18	16, 17	mpan 694	. . . . 5 |- (x e. A -> (F` x) e. ran F)
19	15, 18	eqeltrrd 1546	. . . 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 1709	. 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 954   e. wcel 956  A.wral 1642  Vcvv 1807   (_ wss 2043  {copab 2661  ran crn 3166   Fn wfn 3172  -->wf 3173  ` cfv 3177

This theorem is referenced by:  fopab3 3817  oprcn 7927  ip1cnilem2 8321  ip1cnilem3 8322  ipasslem6 8439  kbass2t 9988

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-fv 3193

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