Theorem txpcofun 5804

Description: Composition distributes over tail cross product in the case of a function. (Contributed by SF, 18-Feb-2015.)

Hypothesis

Ref	Expression
txpcofun.1	⊢ Fun F

Assertion

Ref	Expression
txpcofun	⊢ ((R ⊗ S) ∘ F) = ((R ∘ F) ⊗ (S ∘ F))

Proof of Theorem txpcofun

Dummy variables t x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2863	. . . 4 ⊢ t ∈ V
2		opeqex 4622	. . . 4 ⊢ (t ∈ V → ∃y∃z t = ⟨y, z⟩)
3	1, 2	ax-mp 5	. . 3 ⊢ ∃y∃z t = ⟨y, z⟩
4		dmcoss 4972	. . . . . . . . . 10 ⊢ dom (R ∘ F) ⊆ dom F
5		opeldm 4911	. . . . . . . . . 10 ⊢ (⟨x, y⟩ ∈ (R ∘ F) → x ∈ dom (R ∘ F))
6	4, 5	sseldi 3272	. . . . . . . . 9 ⊢ (⟨x, y⟩ ∈ (R ∘ F) → x ∈ dom F)
7	6	pm4.71ri 614	. . . . . . . 8 ⊢ (⟨x, y⟩ ∈ (R ∘ F) ↔ (x ∈ dom F ∧ ⟨x, y⟩ ∈ (R ∘ F)))
8	7	anbi1i 676	. . . . . . 7 ⊢ ((⟨x, y⟩ ∈ (R ∘ F) ∧ ⟨x, z⟩ ∈ (S ∘ F)) ↔ ((x ∈ dom F ∧ ⟨x, y⟩ ∈ (R ∘ F)) ∧ ⟨x, z⟩ ∈ (S ∘ F)))
9		anass 630	. . . . . . 7 ⊢ (((x ∈ dom F ∧ ⟨x, y⟩ ∈ (R ∘ F)) ∧ ⟨x, z⟩ ∈ (S ∘ F)) ↔ (x ∈ dom F ∧ (⟨x, y⟩ ∈ (R ∘ F) ∧ ⟨x, z⟩ ∈ (S ∘ F))))
10		fvex 5340	. . . . . . . . . . 11 ⊢ (F ‘x) ∈ V
11		breq1 4643	. . . . . . . . . . 11 ⊢ (t = (F ‘x) → (tRy ↔ (F ‘x)Ry))
12	10, 11	ceqsexv 2895	. . . . . . . . . 10 ⊢ (∃t(t = (F ‘x) ∧ tRy) ↔ (F ‘x)Ry)
13		breq1 4643	. . . . . . . . . . 11 ⊢ (t = (F ‘x) → (tSz ↔ (F ‘x)Sz))
14	10, 13	ceqsexv 2895	. . . . . . . . . 10 ⊢ (∃t(t = (F ‘x) ∧ tSz) ↔ (F ‘x)Sz)
15	12, 14	anbi12i 678	. . . . . . . . 9 ⊢ ((∃t(t = (F ‘x) ∧ tRy) ∧ ∃t(t = (F ‘x) ∧ tSz)) ↔ ((F ‘x)Ry ∧ (F ‘x)Sz))
16		eqcom 2355	. . . . . . . . . . . . . 14 ⊢ (t = (F ‘x) ↔ (F ‘x) = t)
17		txpcofun.1	. . . . . . . . . . . . . . 15 ⊢ Fun F
18		funbrfvb 5361	. . . . . . . . . . . . . . 15 ⊢ ((Fun F ∧ x ∈ dom F) → ((F ‘x) = t ↔ xFt))
19	17, 18	mpan 651	. . . . . . . . . . . . . 14 ⊢ (x ∈ dom F → ((F ‘x) = t ↔ xFt))
20	16, 19	syl5bb 248	. . . . . . . . . . . . 13 ⊢ (x ∈ dom F → (t = (F ‘x) ↔ xFt))
21	20	anbi1d 685	. . . . . . . . . . . 12 ⊢ (x ∈ dom F → ((t = (F ‘x) ∧ tRy) ↔ (xFt ∧ tRy)))
22	21	exbidv 1626	. . . . . . . . . . 11 ⊢ (x ∈ dom F → (∃t(t = (F ‘x) ∧ tRy) ↔ ∃t(xFt ∧ tRy)))
23		opelco 4885	. . . . . . . . . . 11 ⊢ (⟨x, y⟩ ∈ (R ∘ F) ↔ ∃t(xFt ∧ tRy))
24	22, 23	syl6bbr 254	. . . . . . . . . 10 ⊢ (x ∈ dom F → (∃t(t = (F ‘x) ∧ tRy) ↔ ⟨x, y⟩ ∈ (R ∘ F)))
25	20	anbi1d 685	. . . . . . . . . . . 12 ⊢ (x ∈ dom F → ((t = (F ‘x) ∧ tSz) ↔ (xFt ∧ tSz)))
26	25	exbidv 1626	. . . . . . . . . . 11 ⊢ (x ∈ dom F → (∃t(t = (F ‘x) ∧ tSz) ↔ ∃t(xFt ∧ tSz)))
27		opelco 4885	. . . . . . . . . . 11 ⊢ (⟨x, z⟩ ∈ (S ∘ F) ↔ ∃t(xFt ∧ tSz))
28	26, 27	syl6bbr 254	. . . . . . . . . 10 ⊢ (x ∈ dom F → (∃t(t = (F ‘x) ∧ tSz) ↔ ⟨x, z⟩ ∈ (S ∘ F)))
29	24, 28	anbi12d 691	. . . . . . . . 9 ⊢ (x ∈ dom F → ((∃t(t = (F ‘x) ∧ tRy) ∧ ∃t(t = (F ‘x) ∧ tSz)) ↔ (⟨x, y⟩ ∈ (R ∘ F) ∧ ⟨x, z⟩ ∈ (S ∘ F))))
30	15, 29	syl5rbbr 251	. . . . . . . 8 ⊢ (x ∈ dom F → ((⟨x, y⟩ ∈ (R ∘ F) ∧ ⟨x, z⟩ ∈ (S ∘ F)) ↔ ((F ‘x)Ry ∧ (F ‘x)Sz)))
31	30	pm5.32i 618	. . . . . . 7 ⊢ ((x ∈ dom F ∧ (⟨x, y⟩ ∈ (R ∘ F) ∧ ⟨x, z⟩ ∈ (S ∘ F))) ↔ (x ∈ dom F ∧ ((F ‘x)Ry ∧ (F ‘x)Sz)))
32	8, 9, 31	3bitrri 263	. . . . . 6 ⊢ ((x ∈ dom F ∧ ((F ‘x)Ry ∧ (F ‘x)Sz)) ↔ (⟨x, y⟩ ∈ (R ∘ F) ∧ ⟨x, z⟩ ∈ (S ∘ F)))
33		opelco 4885	. . . . . . 7 ⊢ (⟨x, ⟨y, z⟩⟩ ∈ ((R ⊗ S) ∘ F) ↔ ∃t(xFt ∧ t(R ⊗ S)⟨y, z⟩))
34		19.41v 1901	. . . . . . . 8 ⊢ (∃t(xFt ∧ ((F ‘x)Ry ∧ (F ‘x)Sz)) ↔ (∃t xFt ∧ ((F ‘x)Ry ∧ (F ‘x)Sz)))
35		funbrfv 5357	. . . . . . . . . . . 12 ⊢ (Fun F → (xFt → (F ‘x) = t))
36	17, 35	ax-mp 5	. . . . . . . . . . 11 ⊢ (xFt → (F ‘x) = t)
37		trtxp 5782	. . . . . . . . . . . 12 ⊢ ((F ‘x)(R ⊗ S)⟨y, z⟩ ↔ ((F ‘x)Ry ∧ (F ‘x)Sz))
38		breq1 4643	. . . . . . . . . . . 12 ⊢ ((F ‘x) = t → ((F ‘x)(R ⊗ S)⟨y, z⟩ ↔ t(R ⊗ S)⟨y, z⟩))
39	37, 38	syl5rbbr 251	. . . . . . . . . . 11 ⊢ ((F ‘x) = t → (t(R ⊗ S)⟨y, z⟩ ↔ ((F ‘x)Ry ∧ (F ‘x)Sz)))
40	36, 39	syl 15	. . . . . . . . . 10 ⊢ (xFt → (t(R ⊗ S)⟨y, z⟩ ↔ ((F ‘x)Ry ∧ (F ‘x)Sz)))
41	40	pm5.32i 618	. . . . . . . . 9 ⊢ ((xFt ∧ t(R ⊗ S)⟨y, z⟩) ↔ (xFt ∧ ((F ‘x)Ry ∧ (F ‘x)Sz)))
42	41	exbii 1582	. . . . . . . 8 ⊢ (∃t(xFt ∧ t(R ⊗ S)⟨y, z⟩) ↔ ∃t(xFt ∧ ((F ‘x)Ry ∧ (F ‘x)Sz)))
43		eldm 4899	. . . . . . . . 9 ⊢ (x ∈ dom F ↔ ∃t xFt)
44	43	anbi1i 676	. . . . . . . 8 ⊢ ((x ∈ dom F ∧ ((F ‘x)Ry ∧ (F ‘x)Sz)) ↔ (∃t xFt ∧ ((F ‘x)Ry ∧ (F ‘x)Sz)))
45	34, 42, 44	3bitr4i 268	. . . . . . 7 ⊢ (∃t(xFt ∧ t(R ⊗ S)⟨y, z⟩) ↔ (x ∈ dom F ∧ ((F ‘x)Ry ∧ (F ‘x)Sz)))
46	33, 45	bitri 240	. . . . . 6 ⊢ (⟨x, ⟨y, z⟩⟩ ∈ ((R ⊗ S) ∘ F) ↔ (x ∈ dom F ∧ ((F ‘x)Ry ∧ (F ‘x)Sz)))
47		oteltxp 5783	. . . . . 6 ⊢ (⟨x, ⟨y, z⟩⟩ ∈ ((R ∘ F) ⊗ (S ∘ F)) ↔ (⟨x, y⟩ ∈ (R ∘ F) ∧ ⟨x, z⟩ ∈ (S ∘ F)))
48	32, 46, 47	3bitr4i 268	. . . . 5 ⊢ (⟨x, ⟨y, z⟩⟩ ∈ ((R ⊗ S) ∘ F) ↔ ⟨x, ⟨y, z⟩⟩ ∈ ((R ∘ F) ⊗ (S ∘ F)))
49		opeq2 4580	. . . . . . 7 ⊢ (t = ⟨y, z⟩ → ⟨x, t⟩ = ⟨x, ⟨y, z⟩⟩)
50	49	eleq1d 2419	. . . . . 6 ⊢ (t = ⟨y, z⟩ → (⟨x, t⟩ ∈ ((R ⊗ S) ∘ F) ↔ ⟨x, ⟨y, z⟩⟩ ∈ ((R ⊗ S) ∘ F)))
51	49	eleq1d 2419	. . . . . 6 ⊢ (t = ⟨y, z⟩ → (⟨x, t⟩ ∈ ((R ∘ F) ⊗ (S ∘ F)) ↔ ⟨x, ⟨y, z⟩⟩ ∈ ((R ∘ F) ⊗ (S ∘ F))))
52	50, 51	bibi12d 312	. . . . 5 ⊢ (t = ⟨y, z⟩ → ((⟨x, t⟩ ∈ ((R ⊗ S) ∘ F) ↔ ⟨x, t⟩ ∈ ((R ∘ F) ⊗ (S ∘ F))) ↔ (⟨x, ⟨y, z⟩⟩ ∈ ((R ⊗ S) ∘ F) ↔ ⟨x, ⟨y, z⟩⟩ ∈ ((R ∘ F) ⊗ (S ∘ F)))))
53	48, 52	mpbiri 224	. . . 4 ⊢ (t = ⟨y, z⟩ → (⟨x, t⟩ ∈ ((R ⊗ S) ∘ F) ↔ ⟨x, t⟩ ∈ ((R ∘ F) ⊗ (S ∘ F))))
54	53	exlimivv 1635	. . 3 ⊢ (∃y∃z t = ⟨y, z⟩ → (⟨x, t⟩ ∈ ((R ⊗ S) ∘ F) ↔ ⟨x, t⟩ ∈ ((R ∘ F) ⊗ (S ∘ F))))
55	3, 54	ax-mp 5	. 2 ⊢ (⟨x, t⟩ ∈ ((R ⊗ S) ∘ F) ↔ ⟨x, t⟩ ∈ ((R ∘ F) ⊗ (S ∘ F)))
56	55	eqrelriv 4851	1 ⊢ ((R ⊗ S) ∘ F) = ((R ∘ F) ⊗ (S ∘ F))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860  ⟨cop 4562   class class class wbr 4640   ∘ ccom 4722  dom cdm 4773  Fun wfun 4776   ‘cfv 4782   ⊗ ctxp 5736

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-1st 4724  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-2nd 4798  df-txp 5737

This theorem is referenced by: (None)