Theorem cores 5085

Description: Restricted first member of a class composition. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 12-Oct-2004.) (Revised by set.mm contributors, 27-Aug-2011.)

Assertion

Ref	Expression
cores	⊢ (ran B ⊆ C → ((A ↾ C) ∘ B) = (A ∘ B))

Proof of Theorem cores

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

Step	Hyp	Ref	Expression
1		brelrn 4961	. . . . . 6 ⊢ (zBy → y ∈ ran B)
2		ssel 3268	. . . . . 6 ⊢ (ran B ⊆ C → (y ∈ ran B → y ∈ C))
3		iba 489	. . . . . . 7 ⊢ (y ∈ C → (yAx ↔ (yAx ∧ y ∈ C)))
4		brres 4950	. . . . . . 7 ⊢ (y(A ↾ C)x ↔ (yAx ∧ y ∈ C))
5	3, 4	syl6rbbr 255	. . . . . 6 ⊢ (y ∈ C → (y(A ↾ C)x ↔ yAx))
6	1, 2, 5	syl56 30	. . . . 5 ⊢ (ran B ⊆ C → (zBy → (y(A ↾ C)x ↔ yAx)))
7	6	pm5.32d 620	. . . 4 ⊢ (ran B ⊆ C → ((zBy ∧ y(A ↾ C)x) ↔ (zBy ∧ yAx)))
8	7	exbidv 1626	. . 3 ⊢ (ran B ⊆ C → (∃y(zBy ∧ y(A ↾ C)x) ↔ ∃y(zBy ∧ yAx)))
9	8	opabbidv 4626	. 2 ⊢ (ran B ⊆ C → {⟨z, x⟩ ∣ ∃y(zBy ∧ y(A ↾ C)x)} = {⟨z, x⟩ ∣ ∃y(zBy ∧ yAx)})
10		df-co 4727	. 2 ⊢ ((A ↾ C) ∘ B) = {⟨z, x⟩ ∣ ∃y(zBy ∧ y(A ↾ C)x)}
11		df-co 4727	. 2 ⊢ (A ∘ B) = {⟨z, x⟩ ∣ ∃y(zBy ∧ yAx)}
12	9, 10, 11	3eqtr4g 2410	1 ⊢ (ran B ⊆ C → ((A ↾ C) ∘ B) = (A ∘ B))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ⊆ wss 3258  {copab 4623   class class class wbr 4640   ∘ ccom 4722  ran crn 4774   ↾ cres 4775

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-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789

This theorem is referenced by:  cores2  5092  fcoi2  5242