Theorem restxp 5786

Description: Restriction distributes over tail cross product. (Contributed by SF, 24-Feb-2015.)

Assertion

Ref	Expression
restxp	⊢ ((A ⊗ B) ↾ C) = ((A ↾ C) ⊗ (B ↾ C))

Proof of Theorem restxp

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

Step	Hyp	Ref	Expression
1		anandir 802	. . . . . 6 ⊢ (((xAa ∧ xBb) ∧ x ∈ C) ↔ ((xAa ∧ x ∈ C) ∧ (xBb ∧ x ∈ C)))
2	1	anbi2i 675	. . . . 5 ⊢ ((y = ⟨a, b⟩ ∧ ((xAa ∧ xBb) ∧ x ∈ C)) ↔ (y = ⟨a, b⟩ ∧ ((xAa ∧ x ∈ C) ∧ (xBb ∧ x ∈ C))))
3		3anass 938	. . . . . . 7 ⊢ ((y = ⟨a, b⟩ ∧ xAa ∧ xBb) ↔ (y = ⟨a, b⟩ ∧ (xAa ∧ xBb)))
4	3	anbi1i 676	. . . . . 6 ⊢ (((y = ⟨a, b⟩ ∧ xAa ∧ xBb) ∧ x ∈ C) ↔ ((y = ⟨a, b⟩ ∧ (xAa ∧ xBb)) ∧ x ∈ C))
5		anass 630	. . . . . 6 ⊢ (((y = ⟨a, b⟩ ∧ (xAa ∧ xBb)) ∧ x ∈ C) ↔ (y = ⟨a, b⟩ ∧ ((xAa ∧ xBb) ∧ x ∈ C)))
6	4, 5	bitri 240	. . . . 5 ⊢ (((y = ⟨a, b⟩ ∧ xAa ∧ xBb) ∧ x ∈ C) ↔ (y = ⟨a, b⟩ ∧ ((xAa ∧ xBb) ∧ x ∈ C)))
7		3anass 938	. . . . 5 ⊢ ((y = ⟨a, b⟩ ∧ (xAa ∧ x ∈ C) ∧ (xBb ∧ x ∈ C)) ↔ (y = ⟨a, b⟩ ∧ ((xAa ∧ x ∈ C) ∧ (xBb ∧ x ∈ C))))
8	2, 6, 7	3bitr4i 268	. . . 4 ⊢ (((y = ⟨a, b⟩ ∧ xAa ∧ xBb) ∧ x ∈ C) ↔ (y = ⟨a, b⟩ ∧ (xAa ∧ x ∈ C) ∧ (xBb ∧ x ∈ C)))
9	8	2exbii 1583	. . 3 ⊢ (∃a∃b((y = ⟨a, b⟩ ∧ xAa ∧ xBb) ∧ x ∈ C) ↔ ∃a∃b(y = ⟨a, b⟩ ∧ (xAa ∧ x ∈ C) ∧ (xBb ∧ x ∈ C)))
10		brtxp 5783	. . . . 5 ⊢ (x(A ⊗ B)y ↔ ∃a∃b(y = ⟨a, b⟩ ∧ xAa ∧ xBb))
11	10	anbi1i 676	. . . 4 ⊢ ((x(A ⊗ B)y ∧ x ∈ C) ↔ (∃a∃b(y = ⟨a, b⟩ ∧ xAa ∧ xBb) ∧ x ∈ C))
12		brres 4949	. . . 4 ⊢ (x((A ⊗ B) ↾ C)y ↔ (x(A ⊗ B)y ∧ x ∈ C))
13		19.41vv 1902	. . . 4 ⊢ (∃a∃b((y = ⟨a, b⟩ ∧ xAa ∧ xBb) ∧ x ∈ C) ↔ (∃a∃b(y = ⟨a, b⟩ ∧ xAa ∧ xBb) ∧ x ∈ C))
14	11, 12, 13	3bitr4i 268	. . 3 ⊢ (x((A ⊗ B) ↾ C)y ↔ ∃a∃b((y = ⟨a, b⟩ ∧ xAa ∧ xBb) ∧ x ∈ C))
15		brtxp 5783	. . . 4 ⊢ (x((A ↾ C) ⊗ (B ↾ C))y ↔ ∃a∃b(y = ⟨a, b⟩ ∧ x(A ↾ C)a ∧ x(B ↾ C)b))
16		biid 227	. . . . . 6 ⊢ (y = ⟨a, b⟩ ↔ y = ⟨a, b⟩)
17		brres 4949	. . . . . 6 ⊢ (x(A ↾ C)a ↔ (xAa ∧ x ∈ C))
18		brres 4949	. . . . . 6 ⊢ (x(B ↾ C)b ↔ (xBb ∧ x ∈ C))
19	16, 17, 18	3anbi123i 1140	. . . . 5 ⊢ ((y = ⟨a, b⟩ ∧ x(A ↾ C)a ∧ x(B ↾ C)b) ↔ (y = ⟨a, b⟩ ∧ (xAa ∧ x ∈ C) ∧ (xBb ∧ x ∈ C)))
20	19	2exbii 1583	. . . 4 ⊢ (∃a∃b(y = ⟨a, b⟩ ∧ x(A ↾ C)a ∧ x(B ↾ C)b) ↔ ∃a∃b(y = ⟨a, b⟩ ∧ (xAa ∧ x ∈ C) ∧ (xBb ∧ x ∈ C)))
21	15, 20	bitri 240	. . 3 ⊢ (x((A ↾ C) ⊗ (B ↾ C))y ↔ ∃a∃b(y = ⟨a, b⟩ ∧ (xAa ∧ x ∈ C) ∧ (xBb ∧ x ∈ C)))
22	9, 14, 21	3bitr4i 268	. 2 ⊢ (x((A ⊗ B) ↾ C)y ↔ x((A ↾ C) ⊗ (B ↾ C))y)
23	22	eqbrriv 4851	1 ⊢ ((A ⊗ B) ↾ C) = ((A ↾ C) ⊗ (B ↾ C))

Syntax hints:   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ⟨cop 4561   class class class wbr 4639   ↾ cres 4774   ⊗ ctxp 5735

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 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-xp 4784  df-cnv 4785  df-res 4788  df-2nd 4797  df-txp 5736

This theorem is referenced by: (None)