Theorem cores2 5092

Description: Absorption of a reverse (preimage) restriction of the second member of a class composition. (Contributed by set.mm contributors, 11-Dec-2006.)

Assertion

Ref	Expression
cores2	⊢ (dom A ⊆ C → (A ∘ ◡(◡B ↾ C)) = (A ∘ B))

Proof of Theorem cores2

Step	Hyp	Ref	Expression
1		cnvcnv 5063	. . . . . 6 ⊢ ◡◡(◡B ↾ C) = (◡B ↾ C)
2	1	coeq1i 4877	. . . . 5 ⊢ (◡◡(◡B ↾ C) ∘ ◡A) = ((◡B ↾ C) ∘ ◡A)
3		df-dm 4788	. . . . . . 7 ⊢ dom A = ran ◡A
4	3	sseq1i 3296	. . . . . 6 ⊢ (dom A ⊆ C ↔ ran ◡A ⊆ C)
5		cores 5085	. . . . . 6 ⊢ (ran ◡A ⊆ C → ((◡B ↾ C) ∘ ◡A) = (◡B ∘ ◡A))
6	4, 5	sylbi 187	. . . . 5 ⊢ (dom A ⊆ C → ((◡B ↾ C) ∘ ◡A) = (◡B ∘ ◡A))
7	2, 6	syl5eq 2397	. . . 4 ⊢ (dom A ⊆ C → (◡◡(◡B ↾ C) ∘ ◡A) = (◡B ∘ ◡A))
8		cnvco 4895	. . . 4 ⊢ ◡(A ∘ ◡(◡B ↾ C)) = (◡◡(◡B ↾ C) ∘ ◡A)
9		cnvco 4895	. . . 4 ⊢ ◡(A ∘ B) = (◡B ∘ ◡A)
10	7, 8, 9	3eqtr4g 2410	. . 3 ⊢ (dom A ⊆ C → ◡(A ∘ ◡(◡B ↾ C)) = ◡(A ∘ B))
11	10	cnveqd 4889	. 2 ⊢ (dom A ⊆ C → ◡◡(A ∘ ◡(◡B ↾ C)) = ◡◡(A ∘ B))
12		cnvcnv 5063	. 2 ⊢ ◡◡(A ∘ ◡(◡B ↾ C)) = (A ∘ ◡(◡B ↾ C))
13		cnvcnv 5063	. 2 ⊢ ◡◡(A ∘ B) = (A ∘ B)
14	11, 12, 13	3eqtr3g 2408	1 ⊢ (dom A ⊆ C → (A ∘ ◡(◡B ↾ C)) = (A ∘ B))

Syntax hints:   → wi 4   = wceq 1642   ⊆ wss 3258   ∘ ccom 4722  ^◡ccnv 4772  dom cdm 4773  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: (None)