Theorem fmpt2co 5779

Description: Composition of two functions. Variation of fmptco 5273 when the second function has two arguments. (Contributed by Mario Carneiro, 8-Feb-2015.)

Hypotheses

Ref	Expression
fmpt2co.1	|- ((ph /\ (x e. A /\ y e. B)) -> R e. C)
fmpt2co.2	|- (ph -> F = (x e. A , y e. B |-> R))
fmpt2co.3	|- (ph -> G = (z e. C |-> S))
fmpt2co.4	|- (z = R -> S = T)

Assertion

Ref	Expression
fmpt2co	|- (ph -> ( G o. F) = (x e. A , y e. B |-> T))

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

Allowed substitution hints:   ph(z)   A(z)   B(z)   R(x,y)   S(z)   T(x,y)   F(x,y,z)   G(x,y,z)

Proof of Theorem fmpt2co

Dummy variables u v w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fmpt2co.1	. . . . . 6 |- ((ph /\ (x e. A /\ y e. B)) -> R e. C)
2	1	ralrimivva 2395	. . . . 5 |- (ph -> A.x e. A A.y e. B R e. C)
3		eqid 2037	. . . . . 6 |- (x e. A , y e. B |-> R) = (x e. A , y e. B |-> R)
4	3	fmpt2 5769	. . . . 5 |- (A.x e. A A.y e. B R e. C <-> (x e. A , y e. B |-> R) :(A X. B) --> C)
5	2, 4	sylib 127	. . . 4 |- (ph -> (x e. A , y e. B |-> R) :(A X. B) --> C)
6		nfcv 2175	. . . . . . 7 |- F/_u R
7		nfcv 2175	. . . . . . 7 |- F/_v R
8		nfcv 2175	. . . . . . . 8 |- F/_xv
9		nfcsb1v 2876	. . . . . . . 8 |- F/_x [_u / x ]_ R
10	8, 9	nfcsb 2878	. . . . . . 7 |- F/_x [_v / y ]_ [_u / x ]_ R
11		nfcsb1v 2876	. . . . . . 7 |- F/_y [_v / y ]_ [_u / x ]_ R
12		csbeq1a 2854	. . . . . . . 8 |- (x = u -> R = [_u / x ]_ R)
13		csbeq1a 2854	. . . . . . . 8 |- (y = v -> [_u / x ]_ R = [_v / y ]_ [_u / x ]_ R)
14	12, 13	sylan9eq 2089	. . . . . . 7 |- ((x = u /\ y = v) -> R = [_v / y ]_ [_u / x ]_ R)
15	6, 7, 10, 11, 14	cbvmpt2 5525	. . . . . 6 |- (x e. A , y e. B |-> R) = (u e. A , v e. B |-> [_v / y ]_ [_u / x ]_ R)
16		vex 2554	. . . . . . . . . 10 |- u e. _V
17		vex 2554	. . . . . . . . . 10 |- v e. _V
18	16, 17	op2ndd 5718	. . . . . . . . 9 |- (w = <.u , v >. -> ( 2nd ` w) = v)
19	18	csbeq1d 2852	. . . . . . . 8 |- (w = <.u , v >. -> [_( 2nd ` w) / y ]_ [_( 1st ` w) / x ]_ R = [_v / y ]_ [_( 1st ` w) / x ]_ R)
20	16, 17	op1std 5717	. . . . . . . . . 10 |- (w = <.u , v >. -> ( 1st ` w) = u)
21	20	csbeq1d 2852	. . . . . . . . 9 |- (w = <.u , v >. -> [_( 1st ` w) / x ]_ R = [_u / x ]_ R)
22	21	csbeq2dv 2869	. . . . . . . 8 |- (w = <.u , v >. -> [_v / y ]_ [_( 1st ` w) / x ]_ R = [_v / y ]_ [_u / x ]_ R)
23	19, 22	eqtrd 2069	. . . . . . 7 |- (w = <.u , v >. -> [_( 2nd ` w) / y ]_ [_( 1st ` w) / x ]_ R = [_v / y ]_ [_u / x ]_ R)
24	23	mpt2mpt 5538	. . . . . 6 |- (w e. (A X. B) |-> [_( 2nd ` w) / y ]_ [_( 1st ` w) / x ]_ R) = (u e. A , v e. B |-> [_v / y ]_ [_u / x ]_ R)
25	15, 24	eqtr4i 2060	. . . . 5 |- (x e. A , y e. B |-> R) = (w e. (A X. B) |-> [_( 2nd ` w) / y ]_ [_( 1st ` w) / x ]_ R)
26	25	fmpt 5262	. . . 4 |- (A.w e. (A X. B) [_( 2nd ` w) / y ]_ [_( 1st ` w) / x ]_ R e. C <-> (x e. A , y e. B |-> R) :(A X. B) --> C)
27	5, 26	sylibr 137	. . 3 |- (ph -> A.w e. (A X. B) [_( 2nd ` w) / y ]_ [_( 1st ` w) / x ]_ R e. C)
28		fmpt2co.2	. . . 4 |- (ph -> F = (x e. A , y e. B |-> R))
29	28, 25	syl6eq 2085	. . 3 |- (ph -> F = (w e. (A X. B) |-> [_( 2nd ` w) / y ]_ [_( 1st ` w) / x ]_ R))
30		fmpt2co.3	. . 3 |- (ph -> G = (z e. C |-> S))
31	27, 29, 30	fmptcos 5275	. 2 |- (ph -> ( G o. F) = (w e. (A X. B) |-> [_ [_( 2nd ` w) / y ]_ [_( 1st ` w) / x ]_ R / z ]_ S))
32	23	csbeq1d 2852	. . . . 5 |- (w = <.u , v >. -> [_ [_( 2nd ` w) / y ]_ [_( 1st ` w) / x ]_ R / z ]_ S = [_ [_v / y ]_ [_u / x ]_ R / z ]_ S)
33	32	mpt2mpt 5538	. . . 4 |- (w e. (A X. B) |-> [_ [_( 2nd ` w) / y ]_ [_( 1st ` w) / x ]_ R / z ]_ S) = (u e. A , v e. B |-> [_ [_v / y ]_ [_u / x ]_ R / z ]_ S)
34		nfcv 2175	. . . . 5 |- F/_u [_ R / z ]_ S
35		nfcv 2175	. . . . 5 |- F/_v [_ R / z ]_ S
36		nfcv 2175	. . . . . 6 |- F/_x S
37	10, 36	nfcsb 2878	. . . . 5 |- F/_x [_ [_v / y ]_ [_u / x ]_ R / z ]_ S
38		nfcv 2175	. . . . . 6 |- F/_y S
39	11, 38	nfcsb 2878	. . . . 5 |- F/_y [_ [_v / y ]_ [_u / x ]_ R / z ]_ S
40	14	csbeq1d 2852	. . . . 5 |- ((x = u /\ y = v) -> [_ R / z ]_ S = [_ [_v / y ]_ [_u / x ]_ R / z ]_ S)
41	34, 35, 37, 39, 40	cbvmpt2 5525	. . . 4 |- (x e. A , y e. B |-> [_ R / z ]_ S) = (u e. A , v e. B |-> [_ [_v / y ]_ [_u / x ]_ R / z ]_ S)
42	33, 41	eqtr4i 2060	. . 3 |- (w e. (A X. B) |-> [_ [_( 2nd ` w) / y ]_ [_( 1st ` w) / x ]_ R / z ]_ S) = (x e. A , y e. B |-> [_ R / z ]_ S)
43	1	3impb 1099	. . . . 5 |- ((ph /\ x e. A /\ y e. B) -> R e. C)
44		nfcvd 2176	. . . . . 6 |- ( R e. C -> F/_z T)
45		fmpt2co.4	. . . . . 6 |- (z = R -> S = T)
46	44, 45	csbiegf 2884	. . . . 5 |- ( R e. C -> [_ R / z ]_ S = T)
47	43, 46	syl 14	. . . 4 |- ((ph /\ x e. A /\ y e. B) -> [_ R / z ]_ S = T)
48	47	mpt2eq3dva 5511	. . 3 |- (ph -> (x e. A , y e. B |-> [_ R / z ]_ S) = (x e. A , y e. B |-> T))
49	42, 48	syl5eq 2081	. 2 |- (ph -> (w e. (A X. B) |-> [_ [_( 2nd ` w) / y ]_ [_( 1st ` w) / x ]_ R / z ]_ S) = (x e. A , y e. B |-> T))
50	31, 49	eqtrd 2069	1 |- (ph -> ( G o. F) = (x e. A , y e. B |-> T))

Syntax hints:   -> wi 4   /\ wa 97   /\ w3a 884   = wceq 1242   e. wcel 1390  A.wral 2300   [_csb 2846   <.cop 3370   |-> cmpt 3809   X. cxp 4286   o. ccom 4292   -->wf 4841   ` cfv 4845   |-> cmpt2 5457   1stc1st 5707   2ndc2nd 5708

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fv 4853  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710

This theorem is referenced by:  oprabco  5780