Theorem ov 5596

Description: The value of an operation class abstraction. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by set.mm contributors, 16-May-1995.) (Revised by set.mm contributors, 24-Jul-2012.)

Hypotheses

Ref	Expression
ov.1	|- C e. _V
ov.2	|- (x = A -> (ph <-> ps))
ov.3	|- (y = B -> (ps <-> ch))
ov.4	|- (z = C -> (ch <-> th))
ov.5	|- ((x e. R /\ y e. S) -> E!zph)
ov.6	|- F = {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}

Assertion

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

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

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

Proof of Theorem ov

Step	Hyp	Ref	Expression
1		df-ov 5527	. . . . 5 |- (AFB) = (F` <.A, B>.)
2		ov.6	. . . . . 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		ov.5	. . . . . 6 |- ((x e. R /\ y e. S) -> E!zph)
7	6	fnoprab 5587	. . . . 5 |- {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)} Fn {<.x, y>. | (x e. R /\ y e. S)}
8		eleq1 2413	. . . . . . . 8 |- (x = A -> (x e. R <-> A e. R))
9	8	anbi1d 685	. . . . . . 7 |- (x = A -> ((x e. R /\ y e. S) <-> (A e. R /\ y e. S)))
10		eleq1 2413	. . . . . . . 8 |- (y = B -> (y e. S <-> B e. S))
11	10	anbi2d 684	. . . . . . 7 |- (y = B -> ((A e. R /\ y e. S) <-> (A e. R /\ B e. S)))
12	9, 11	opelopabg 4706	. . . . . 6 |- ((A e. R /\ B e. S) -> (<.A, B>. e. {<.x, y>. | (x e. R /\ y e. S)} <-> (A e. R /\ B e. S)))
13	12	ibir 233	. . . . 5 |- ((A e. R /\ B e. S) -> <.A, B>. e. {<.x, y>. | (x e. R /\ y e. S)})
14		fnopfvb 5360	. . . . 5 |- (({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)} Fn {<.x, y>. | (x e. R /\ y e. S)} /\ <.A, B>. e. {<.x, y>. | (x e. R /\ y e. 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)}))
15	7, 13, 14	sylancr 644	. . . 4 |- ((A e. R /\ B e. 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)}))
16		ov.1	. . . . 5 |- C e. _V
17		ov.2	. . . . . . 7 |- (x = A -> (ph <-> ps))
18	9, 17	anbi12d 691	. . . . . 6 |- (x = A -> (((x e. R /\ y e. S) /\ ph) <-> ((A e. R /\ y e. S) /\ ps)))
19		ov.3	. . . . . . 7 |- (y = B -> (ps <-> ch))
20	11, 19	anbi12d 691	. . . . . 6 |- (y = B -> (((A e. R /\ y e. S) /\ ps) <-> ((A e. R /\ B e. S) /\ ch)))
21		ov.4	. . . . . . 7 |- (z = C -> (ch <-> th))
22	21	anbi2d 684	. . . . . 6 |- (z = C -> (((A e. R /\ B e. S) /\ ch) <-> ((A e. R /\ B e. S) /\ th)))
23	18, 20, 22	eloprabg 5580	. . . . 5 |- ((A e. R /\ B e. S /\ C e. _V) -> (<.<.A, B>., C>. e. {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)} <-> ((A e. R /\ B e. S) /\ th)))
24	16, 23	mp3an3 1266	. . . 4 |- ((A e. R /\ B e. S) -> (<.<.A, B>., C>. e. {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)} <-> ((A e. R /\ B e. S) /\ th)))
25	15, 24	bitrd 244	. . 3 |- ((A e. R /\ B e. S) -> (({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.) = C <-> ((A e. R /\ B e. S) /\ th)))
26	5, 25	syl5bb 248	. 2 |- ((A e. R /\ B e. S) -> ((AFB) = C <-> ((A e. R /\ B e. S) /\ th)))
27	26	bianabs 850	1 |- ((A e. R /\ B e. S) -> ((AFB) = C <-> th))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710  E!weu 2204  _Vcvv 2860  <.cop 4562  {copab 4623   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-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)