Theorem oprabex2gpop 14337

Description: Existence of an operation class abstraction. (A version of oprabex2g 4949 adapted to partial operations.)

Assertion

Ref	Expression
oprabex2gpop	|- ((R e. B /\ Rel R) -> {<.<.x, y>., z>. | (<.x, y>. e. R /\ z = A)} e. _V)

Distinct variable groups:   z,A   x,B,y,z   x,R,y,z

Proof of Theorem oprabex2gpop

Step	Hyp	Ref	Expression
1		relssdmrn 4416	. . . . . 6 |- (Rel R -> R C_ (dom R X. ran R))
2	1	adantl 424	. . . . 5 |- ((R e. B /\ Rel R) -> R C_ (dom R X. ran R))
3	2	sseld 2619	. . . 4 |- ((R e. B /\ Rel R) -> (<.x, y>. e. R -> <.x, y>. e. (dom R X. ran R)))
4	3	anim1d 619	. . 3 |- ((R e. B /\ Rel R) -> ((<.x, y>. e. R /\ z = A) -> (<.x, y>. e. (dom R X. ran R) /\ z = A)))
5	4	ssoprab2g 14333	. 2 |- ((R e. B /\ Rel R) -> {<.<.x, y>., z>. | (<.x, y>. e. R /\ z = A)} C_ {<.<.x, y>., z>. | (<.x, y>. e. (dom R X. ran R) /\ z = A)})
6		dmexg 4206	. . . . . 6 |- (R e. B -> dom R e. _V)
7		rnexg 4207	. . . . . 6 |- (R e. B -> ran R e. _V)
8	6, 7	jca 310	. . . . 5 |- (R e. B -> (dom R e. _V /\ ran R e. _V))
9	8	adantr 425	. . . 4 |- ((R e. B /\ Rel R) -> (dom R e. _V /\ ran R e. _V))
10		eqid 1884	. . . . 5 |- {<.<.x, y>., z>. | ((x e. dom R /\ y e. ran R) /\ z = A)} = {<.<.x, y>., z>. | ((x e. dom R /\ y e. ran R) /\ z = A)}
11	10	oprabex2g 4949	. . . 4 |- ((dom R e. _V /\ ran R e. _V) -> {<.<.x, y>., z>. | ((x e. dom R /\ y e. ran R) /\ z = A)} e. _V)
12	9, 11	syl 12	. . 3 |- ((R e. B /\ Rel R) -> {<.<.x, y>., z>. | ((x e. dom R /\ y e. ran R) /\ z = A)} e. _V)
13		twsvbdop 14332	. . 3 |- {<.<.x, y>., z>. | (<.x, y>. e. (dom R X. ran R) /\ z = A)} = {<.<.x, y>., z>. | ((x e. dom R /\ y e. ran R) /\ z = A)}
14	12, 13	syl5eqel 1975	. 2 |- ((R e. B /\ Rel R) -> {<.<.x, y>., z>. | (<.x, y>. e. (dom R X. ran R) /\ z = A)} e. _V)
15		ssexg 3457	. 2 |- (({<.<.x, y>., z>. | (<.x, y>. e. R /\ z = A)} C_ {<.<.x, y>., z>. | (<.x, y>. e. (dom R X. ran R) /\ z = A)} /\ {<.<.x, y>., z>. | (<.x, y>. e. (dom R X. ran R) /\ z = A)} e. _V) -> {<.<.x, y>., z>. | (<.x, y>. e. R /\ z = A)} e. _V)
16	5, 14, 15	syl11anc 524	1 |- ((R e. B /\ Rel R) -> {<.<.x, y>., z>. | (<.x, y>. e. R /\ z = A)} e. _V)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292   C_ wss 2593  <.cop 3046   X. cxp 3984  dom cdm 3986  ran crn 3987  Rel wrel 3991  {copab2 4885

This theorem is referenced by:  dualalg 15131  dualded 15132

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-rep 3428  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-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-oprab 4887

Copyright terms: Public domain