Theorem dfoprab4spop 14339

Description: Class abstraction for operations in terms of class abstraction of ordered pairs. (A version of dfoprab4s 5056 adapted to partial operations.)

Hypothesis

Ref	Expression
dfoprab4spop.1	|- (<.x, y>. = w -> (ph <-> ps))

Assertion

Ref	Expression
dfoprab4spop	|- (Rel R -> {<.<.x, y>., z>. | (<.x, y>. e. R /\ ph)} = {<.w, z>. | (w e. R /\ ps)})

Distinct variable groups:   w,R,x,y,z   ph,w   ps,x,y

Proof of Theorem dfoprab4spop

Step	Hyp	Ref	Expression
1		df-rel 4001	. . . . . . 7 |- (Rel R <-> R C_ (_V X. _V))
2		df-ss 2605	. . . . . . . 8 |- (R C_ (_V X. _V) <-> (R i^i (_V X. _V)) = R)
3		incom 2787	. . . . . . . . 9 |- ((_V X. _V) i^i R) = (R i^i (_V X. _V))
4		eqtr 1904	. . . . . . . . . 10 |- ((((_V X. _V) i^i R) = (R i^i (_V X. _V)) /\ (R i^i (_V X. _V)) = R) -> ((_V X. _V) i^i R) = R)
5	4	eleq2d 1964	. . . . . . . . 9 |- ((((_V X. _V) i^i R) = (R i^i (_V X. _V)) /\ (R i^i (_V X. _V)) = R) -> (w e. ((_V X. _V) i^i R) <-> w e. R))
6	3, 5	mpan 759	. . . . . . . 8 |- ((R i^i (_V X. _V)) = R -> (w e. ((_V X. _V) i^i R) <-> w e. R))
7	2, 6	sylbi 216	. . . . . . 7 |- (R C_ (_V X. _V) -> (w e. ((_V X. _V) i^i R) <-> w e. R))
8	1, 7	sylbi 216	. . . . . 6 |- (Rel R -> (w e. ((_V X. _V) i^i R) <-> w e. R))
9		elin 2786	. . . . . 6 |- (w e. ((_V X. _V) i^i R) <-> (w e. (_V X. _V) /\ w e. R))
10	8, 9	syl5bbr 593	. . . . 5 |- (Rel R -> ((w e. (_V X. _V) /\ w e. R) <-> w e. R))
11	10	anbi1d 679	. . . 4 |- (Rel R -> (((w e. (_V X. _V) /\ w e. R) /\ ps) <-> (w e. R /\ ps)))
12		anass 487	. . . 4 |- (((w e. (_V X. _V) /\ w e. R) /\ ps) <-> (w e. (_V X. _V) /\ (w e. R /\ ps)))
13	11, 12	syl5bbr 593	. . 3 |- (Rel R -> ((w e. (_V X. _V) /\ (w e. R /\ ps)) <-> (w e. R /\ ps)))
14	13	opabbidv 3401	. 2 |- (Rel R -> {<.w, z>. | (w e. (_V X. _V) /\ (w e. R /\ ps))} = {<.w, z>. | (w e. R /\ ps)})
15		eleq1 1957	. . . 4 |- (<.x, y>. = w -> (<.x, y>. e. R <-> w e. R))
16		dfoprab4spop.1	. . . 4 |- (<.x, y>. = w -> (ph <-> ps))
17	15, 16	anbi12d 690	. . 3 |- (<.x, y>. = w -> ((<.x, y>. e. R /\ ph) <-> (w e. R /\ ps)))
18	17	dfoprab3s 5055	. 2 |- {<.<.x, y>., z>. | (<.x, y>. e. R /\ ph)} = {<.w, z>. | (w e. (_V X. _V) /\ (w e. R /\ ps))}
19	14, 18	syl5eq 1940	1 |- (Rel R -> {<.<.x, y>., z>. | (<.x, y>. e. R /\ ph)} = {<.w, z>. | (w e. R /\ ps)})

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292   i^i cin 2592   C_ wss 2593  <.cop 3046  {copab 3395   X. cxp 3984  Rel wrel 3991  {copab2 4885

This theorem is referenced by:  dualalg 15131

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-3an 860  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-sbc 2454  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-fv 4014  df-oprab 4887  df-1st 5020  df-2nd 5021

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