Theorem ovg 5602

Description: The value of an operation class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)

Hypotheses

Ref	Expression
ovg.1	|- (x = A -> (ph <-> ps))
ovg.2	|- (y = B -> (ps <-> ch))
ovg.3	|- (z = C -> (ch <-> th))
ovg.4	|- ((ta /\ (x e. R /\ y e. S)) -> E!zph)
ovg.5	|- F = {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}

Assertion

Ref	Expression
ovg	|- ((ta /\ (A e. R /\ B e. S /\ C e. D)) -> ((AFB) = C <-> th))

Distinct variable groups:   ps,x   ch,x,y   th,x,y,z   ta,x,y   x,R,y,z   x,S,y,z   x,A,y,z   x,B,y,z   x,C,y,z

Allowed substitution hints:   ph(x,y,z)   ps(y,z)   ch(z)   ta(z)   D(x,y,z)   F(x,y,z)

Proof of Theorem ovg

Step	Hyp	Ref	Expression
1		df-ov 5527	. . . . 5 |- (AFB) = (F` <.A, B>.)
2		ovg.5	. . . . . 6 |- F = {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}
3	2	fveq1i 5330	. . . . 5 |- (F` <.A, B>.) = ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.)
4	1, 3	eqtri 2373	. . . 4 |- (AFB) = ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.)
5	4	eqeq1i 2360	. . 3 |- ((AFB) = C <-> ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.) = C)
6		ovg.4	. . . . . . . . 9 |- ((ta /\ (x e. R /\ y e. S)) -> E!zph)
7	6	ex 423	. . . . . . . 8 |- (ta -> ((x e. R /\ y e. S) -> E!zph))
8	7	alrimivv 1632	. . . . . . 7 |- (ta -> A.xA.y((x e. R /\ y e. S) -> E!zph))
9		fnoprabg 5586	. . . . . . 7 |- (A.xA.y((x e. R /\ y e. S) -> E!zph) -> {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)} Fn {<.x, y>. | (x e. R /\ y e. S)})
10	8, 9	syl 15	. . . . . 6 |- (ta -> {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)} Fn {<.x, y>. | (x e. R /\ y e. S)})
11		df-xp 4785	. . . . . . 7 |- (R X. S) = {<.x, y>. | (x e. R /\ y e. S)}
12	11	fneq2i 5180	. . . . . 6 |- ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)} Fn (R X. S) <-> {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)} Fn {<.x, y>. | (x e. R /\ y e. S)})
13	10, 12	sylibr 203	. . . . 5 |- (ta -> {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)} Fn (R X. S))
14		opelxp 4812	. . . . . . 7 |- (<.A, B>. e. (R X. S) <-> (A e. R /\ B e. S))
15	14	biimpri 197	. . . . . 6 |- ((A e. R /\ B e. S) -> <.A, B>. e. (R X. S))
16	15	3adant3 975	. . . . 5 |- ((A e. R /\ B e. S /\ C e. D) -> <.A, B>. e. (R X. S))
17		fnopfvb 5360	. . . . 5 |- (({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)} Fn (R X. S) /\ <.A, B>. e. (R X. S)) -> (({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.) = C <-> <.<.A, B>., C>. e. {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}))
18	13, 16, 17	syl2an 463	. . . 4 |- ((ta /\ (A e. R /\ B e. S /\ C e. D)) -> (({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.) = C <-> <.<.A, B>., C>. e. {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}))
19		eleq1 2413	. . . . . . . 8 |- (x = A -> (x e. R <-> A e. R))
20	19	anbi1d 685	. . . . . . 7 |- (x = A -> ((x e. R /\ y e. S) <-> (A e. R /\ y e. S)))
21		ovg.1	. . . . . . 7 |- (x = A -> (ph <-> ps))
22	20, 21	anbi12d 691	. . . . . 6 |- (x = A -> (((x e. R /\ y e. S) /\ ph) <-> ((A e. R /\ y e. S) /\ ps)))
23		eleq1 2413	. . . . . . . 8 |- (y = B -> (y e. S <-> B e. S))
24	23	anbi2d 684	. . . . . . 7 |- (y = B -> ((A e. R /\ y e. S) <-> (A e. R /\ B e. S)))
25		ovg.2	. . . . . . 7 |- (y = B -> (ps <-> ch))
26	24, 25	anbi12d 691	. . . . . 6 |- (y = B -> (((A e. R /\ y e. S) /\ ps) <-> ((A e. R /\ B e. S) /\ ch)))
27		ovg.3	. . . . . . 7 |- (z = C -> (ch <-> th))
28	27	anbi2d 684	. . . . . 6 |- (z = C -> (((A e. R /\ B e. S) /\ ch) <-> ((A e. R /\ B e. S) /\ th)))
29	22, 26, 28	eloprabg 5580	. . . . 5 |- ((A e. R /\ B e. S /\ C e. D) -> (<.<.A, B>., C>. e. {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)} <-> ((A e. R /\ B e. S) /\ th)))
30	29	adantl 452	. . . 4 |- ((ta /\ (A e. R /\ B e. S /\ C e. D)) -> (<.<.A, B>., C>. e. {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)} <-> ((A e. R /\ B e. S) /\ th)))
31	18, 30	bitrd 244	. . 3 |- ((ta /\ (A e. R /\ B e. S /\ C e. D)) -> (({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.) = C <-> ((A e. R /\ B e. S) /\ th)))
32	5, 31	syl5bb 248	. 2 |- ((ta /\ (A e. R /\ B e. S /\ C e. D)) -> ((AFB) = C <-> ((A e. R /\ B e. S) /\ th)))
33		biidd 228	. . . . 5 |- ((A e. R /\ B e. S) -> (((A e. R /\ B e. S) /\ th) <-> ((A e. R /\ B e. S) /\ th)))
34	33	bianabs 850	. . . 4 |- ((A e. R /\ B e. S) -> (((A e. R /\ B e. S) /\ th) <-> th))
35	34	3adant3 975	. . 3 |- ((A e. R /\ B e. S /\ C e. D) -> (((A e. R /\ B e. S) /\ th) <-> th))
36	35	adantl 452	. 2 |- ((ta /\ (A e. R /\ B e. S /\ C e. D)) -> (((A e. R /\ B e. S) /\ th) <-> th))
37	32, 36	bitrd 244	1 |- ((ta /\ (A e. R /\ B e. S /\ C e. D)) -> ((AFB) = C <-> th))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934  A.wal 1540   = wceq 1642   e. wcel 1710  E!weu 2204  <.cop 4562  {copab 4623   X. cxp 4771   Fn wfn 4777  ` cfv 4782  (class class class)co 5526  {coprab 5528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-fv 4796  df-ov 5527  df-oprab 5529

This theorem is referenced by: (None)