Theorem fntxp 5804

Description: If F and G are functions, then their tail cross product is a function over the intersection of their domains. (Contributed by SF, 24-Feb-2015.)

Assertion

Ref	Expression
fntxp	|- ((F Fn A /\ G Fn B) -> (F (x) G) Fn (A i^i B))

Proof of Theorem fntxp

Dummy variables a b c d x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brtxp 5783	. . . . . . . . . 10 |- (x(F (x) G)y <-> E.aE.b(y = <.a, b>. /\ xFa /\ xGb))
2		brtxp 5783	. . . . . . . . . 10 |- (x(F (x) G)z <-> E.cE.d(z = <.c, d>. /\ xFc /\ xGd))
3	1, 2	anbi12i 678	. . . . . . . . 9 |- ((x(F (x) G)y /\ x(F (x) G)z) <-> (E.aE.b(y = <.a, b>. /\ xFa /\ xGb) /\ E.cE.d(z = <.c, d>. /\ xFc /\ xGd)))
4		ee4anv 1915	. . . . . . . . 9 |- (E.aE.bE.cE.d((y = <.a, b>. /\ xFa /\ xGb) /\ (z = <.c, d>. /\ xFc /\ xGd)) <-> (E.aE.b(y = <.a, b>. /\ xFa /\ xGb) /\ E.cE.d(z = <.c, d>. /\ xFc /\ xGd)))
5	3, 4	bitr4i 243	. . . . . . . 8 |- ((x(F (x) G)y /\ x(F (x) G)z) <-> E.aE.bE.cE.d((y = <.a, b>. /\ xFa /\ xGb) /\ (z = <.c, d>. /\ xFc /\ xGd)))
6		an6 1261	. . . . . . . . . . 11 |- (((y = <.a, b>. /\ xFa /\ xGb) /\ (z = <.c, d>. /\ xFc /\ xGd)) <-> ((y = <.a, b>. /\ z = <.c, d>.) /\ (xFa /\ xFc) /\ (xGb /\ xGd)))
7		fununiq 5517	. . . . . . . . . . . . . . . 16 |- ((Fun F /\ xFa /\ xFc) -> a = c)
8	7	3expib 1154	. . . . . . . . . . . . . . 15 |- (Fun F -> ((xFa /\ xFc) -> a = c))
9		fununiq 5517	. . . . . . . . . . . . . . . 16 |- ((Fun G /\ xGb /\ xGd) -> b = d)
10	9	3expib 1154	. . . . . . . . . . . . . . 15 |- (Fun G -> ((xGb /\ xGd) -> b = d))
11	8, 10	im2anan9 808	. . . . . . . . . . . . . 14 |- ((Fun F /\ Fun G) -> (((xFa /\ xFc) /\ (xGb /\ xGd)) -> (a = c /\ b = d)))
12		eqeq12 2365	. . . . . . . . . . . . . . . 16 |- ((y = <.a, b>. /\ z = <.c, d>.) -> (y = z <-> <.a, b>. = <.c, d>.))
13		opth 4602	. . . . . . . . . . . . . . . 16 |- (<.a, b>. = <.c, d>. <-> (a = c /\ b = d))
14	12, 13	syl6bb 252	. . . . . . . . . . . . . . 15 |- ((y = <.a, b>. /\ z = <.c, d>.) -> (y = z <-> (a = c /\ b = d)))
15	14	imbi2d 307	. . . . . . . . . . . . . 14 |- ((y = <.a, b>. /\ z = <.c, d>.) -> ((((xFa /\ xFc) /\ (xGb /\ xGd)) -> y = z) <-> (((xFa /\ xFc) /\ (xGb /\ xGd)) -> (a = c /\ b = d))))
16	11, 15	syl5ibrcom 213	. . . . . . . . . . . . 13 |- ((Fun F /\ Fun G) -> ((y = <.a, b>. /\ z = <.c, d>.) -> (((xFa /\ xFc) /\ (xGb /\ xGd)) -> y = z)))
17	16	exp4a 589	. . . . . . . . . . . 12 |- ((Fun F /\ Fun G) -> ((y = <.a, b>. /\ z = <.c, d>.) -> ((xFa /\ xFc) -> ((xGb /\ xGd) -> y = z))))
18	17	3impd 1165	. . . . . . . . . . 11 |- ((Fun F /\ Fun G) -> (((y = <.a, b>. /\ z = <.c, d>.) /\ (xFa /\ xFc) /\ (xGb /\ xGd)) -> y = z))
19	6, 18	syl5bi 208	. . . . . . . . . 10 |- ((Fun F /\ Fun G) -> (((y = <.a, b>. /\ xFa /\ xGb) /\ (z = <.c, d>. /\ xFc /\ xGd)) -> y = z))
20	19	exlimdvv 1637	. . . . . . . . 9 |- ((Fun F /\ Fun G) -> (E.cE.d((y = <.a, b>. /\ xFa /\ xGb) /\ (z = <.c, d>. /\ xFc /\ xGd)) -> y = z))
21	20	exlimdvv 1637	. . . . . . . 8 |- ((Fun F /\ Fun G) -> (E.aE.bE.cE.d((y = <.a, b>. /\ xFa /\ xGb) /\ (z = <.c, d>. /\ xFc /\ xGd)) -> y = z))
22	5, 21	syl5bi 208	. . . . . . 7 |- ((Fun F /\ Fun G) -> ((x(F (x) G)y /\ x(F (x) G)z) -> y = z))
23	22	alrimiv 1631	. . . . . 6 |- ((Fun F /\ Fun G) -> A.z((x(F (x) G)y /\ x(F (x) G)z) -> y = z))
24	23	alrimivv 1632	. . . . 5 |- ((Fun F /\ Fun G) -> A.xA.yA.z((x(F (x) G)y /\ x(F (x) G)z) -> y = z))
25		dffun2 5119	. . . . 5 |- (Fun (F (x) G) <-> A.xA.yA.z((x(F (x) G)y /\ x(F (x) G)z) -> y = z))
26	24, 25	sylibr 203	. . . 4 |- ((Fun F /\ Fun G) -> Fun (F (x) G))
27		dmtxp 5802	. . . . 5 |- dom (F (x) G) = (dom F i^i dom G)
28		ineq12 3452	. . . . 5 |- ((dom F = A /\ dom G = B) -> (dom F i^i dom G) = (A i^i B))
29	27, 28	syl5eq 2397	. . . 4 |- ((dom F = A /\ dom G = B) -> dom (F (x) G) = (A i^i B))
30	26, 29	anim12i 549	. . 3 |- (((Fun F /\ Fun G) /\ (dom F = A /\ dom G = B)) -> (Fun (F (x) G) /\ dom (F (x) G) = (A i^i B)))
31	30	an4s 799	. 2 |- (((Fun F /\ dom F = A) /\ (Fun G /\ dom G = B)) -> (Fun (F (x) G) /\ dom (F (x) G) = (A i^i B)))
32		df-fn 4790	. . 3 |- (F Fn A <-> (Fun F /\ dom F = A))
33		df-fn 4790	. . 3 |- (G Fn B <-> (Fun G /\ dom G = B))
34	32, 33	anbi12i 678	. 2 |- ((F Fn A /\ G Fn B) <-> ((Fun F /\ dom F = A) /\ (Fun G /\ dom G = B)))
35		df-fn 4790	. 2 |- ((F (x) G) Fn (A i^i B) <-> (Fun (F (x) G) /\ dom (F (x) G) = (A i^i B)))
36	31, 34, 35	3imtr4i 257	1 |- ((F Fn A /\ G Fn B) -> (F (x) G) Fn (A i^i B))

Syntax hints:   -> wi 4   /\ wa 358   /\ w3a 934  A.wal 1540  E.wex 1541   = wceq 1642   i^i cin 3208  <.cop 4561   class class class wbr 4639  dom cdm 4772  Fun wfun 4775   Fn wfn 4776   (x) ctxp 5735

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-co 4726  df-ima 4727  df-id 4767  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-2nd 4797  df-txp 5736

This theorem is referenced by:  xpassen  6057