Theorem dmcosseq 4973

Description: Domain of a composition. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 28-May-1998.) (Revised by set.mm contributors, 27-Aug-2011.)

Assertion

Ref	Expression
dmcosseq	⊢ (ran B ⊆ dom A → dom (A ∘ B) = dom B)

Proof of Theorem dmcosseq

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

Step	Hyp	Ref	Expression
1		dmcoss 4971	. . 3 ⊢ dom (A ∘ B) ⊆ dom B
2	1	a1i 10	. 2 ⊢ (ran B ⊆ dom A → dom (A ∘ B) ⊆ dom B)
3		brelrn 4960	. . . . . . . . . 10 ⊢ (xBy → y ∈ ran B)
4		ssel 3267	. . . . . . . . . 10 ⊢ (ran B ⊆ dom A → (y ∈ ran B → y ∈ dom A))
5	3, 4	syl5 28	. . . . . . . . 9 ⊢ (ran B ⊆ dom A → (xBy → y ∈ dom A))
6		eldm 4898	. . . . . . . . 9 ⊢ (y ∈ dom A ↔ ∃z yAz)
7	5, 6	syl6ib 217	. . . . . . . 8 ⊢ (ran B ⊆ dom A → (xBy → ∃z yAz))
8	7	ancld 536	. . . . . . 7 ⊢ (ran B ⊆ dom A → (xBy → (xBy ∧ ∃z yAz)))
9		19.42v 1905	. . . . . . 7 ⊢ (∃z(xBy ∧ yAz) ↔ (xBy ∧ ∃z yAz))
10	8, 9	syl6ibr 218	. . . . . 6 ⊢ (ran B ⊆ dom A → (xBy → ∃z(xBy ∧ yAz)))
11	10	eximdv 1622	. . . . 5 ⊢ (ran B ⊆ dom A → (∃y xBy → ∃y∃z(xBy ∧ yAz)))
12		brco 4883	. . . . . . 7 ⊢ (x(A ∘ B)z ↔ ∃y(xBy ∧ yAz))
13	12	exbii 1582	. . . . . 6 ⊢ (∃z x(A ∘ B)z ↔ ∃z∃y(xBy ∧ yAz))
14		excom 1741	. . . . . 6 ⊢ (∃z∃y(xBy ∧ yAz) ↔ ∃y∃z(xBy ∧ yAz))
15	13, 14	bitri 240	. . . . 5 ⊢ (∃z x(A ∘ B)z ↔ ∃y∃z(xBy ∧ yAz))
16	11, 15	syl6ibr 218	. . . 4 ⊢ (ran B ⊆ dom A → (∃y xBy → ∃z x(A ∘ B)z))
17		eldm 4898	. . . 4 ⊢ (x ∈ dom B ↔ ∃y xBy)
18		eldm 4898	. . . 4 ⊢ (x ∈ dom (A ∘ B) ↔ ∃z x(A ∘ B)z)
19	16, 17, 18	3imtr4g 261	. . 3 ⊢ (ran B ⊆ dom A → (x ∈ dom B → x ∈ dom (A ∘ B)))
20	19	ssrdv 3278	. 2 ⊢ (ran B ⊆ dom A → dom B ⊆ dom (A ∘ B))
21	2, 20	eqssd 3289	1 ⊢ (ran B ⊆ dom A → dom (A ∘ B) = dom B)

Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ⊆ wss 3257   class class class wbr 4639   ∘ ccom 4721  dom cdm 4772  ran crn 4773

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-co 4726  df-ima 4727  df-cnv 4785  df-rn 4786  df-dm 4787

This theorem is referenced by:  dmcoeq  4974  fnco  5191  sbthlem3  6205