Theorem rnco 5088

Description: The range of the composition of two classes. (Contributed by set.mm contributors, 12-Dec-2006.)

Assertion

Ref	Expression
rnco	⊢ ran (A ∘ B) = ran (A ↾ ran B)

Proof of Theorem rnco

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

Step	Hyp	Ref	Expression
1		brco 4884	. . . . 5 ⊢ (x(A ∘ B)y ↔ ∃z(xBz ∧ zAy))
2	1	exbii 1582	. . . 4 ⊢ (∃x x(A ∘ B)y ↔ ∃x∃z(xBz ∧ zAy))
3		excom 1741	. . . 4 ⊢ (∃x∃z(xBz ∧ zAy) ↔ ∃z∃x(xBz ∧ zAy))
4		ancom 437	. . . . . . 7 ⊢ ((∃x xBz ∧ zAy) ↔ (zAy ∧ ∃x xBz))
5		19.41v 1901	. . . . . . 7 ⊢ (∃x(xBz ∧ zAy) ↔ (∃x xBz ∧ zAy))
6		elrn 4897	. . . . . . . 8 ⊢ (z ∈ ran B ↔ ∃x xBz)
7	6	anbi2i 675	. . . . . . 7 ⊢ ((zAy ∧ z ∈ ran B) ↔ (zAy ∧ ∃x xBz))
8	4, 5, 7	3bitr4i 268	. . . . . 6 ⊢ (∃x(xBz ∧ zAy) ↔ (zAy ∧ z ∈ ran B))
9		brres 4950	. . . . . 6 ⊢ (z(A ↾ ran B)y ↔ (zAy ∧ z ∈ ran B))
10	8, 9	bitr4i 243	. . . . 5 ⊢ (∃x(xBz ∧ zAy) ↔ z(A ↾ ran B)y)
11	10	exbii 1582	. . . 4 ⊢ (∃z∃x(xBz ∧ zAy) ↔ ∃z z(A ↾ ran B)y)
12	2, 3, 11	3bitri 262	. . 3 ⊢ (∃x x(A ∘ B)y ↔ ∃z z(A ↾ ran B)y)
13		elrn 4897	. . 3 ⊢ (y ∈ ran (A ∘ B) ↔ ∃x x(A ∘ B)y)
14		elrn 4897	. . 3 ⊢ (y ∈ ran (A ↾ ran B) ↔ ∃z z(A ↾ ran B)y)
15	12, 13, 14	3bitr4i 268	. 2 ⊢ (y ∈ ran (A ∘ B) ↔ y ∈ ran (A ↾ ran B))
16	15	eqriv 2350	1 ⊢ ran (A ∘ B) = ran (A ↾ ran B)

Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710   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-rn 4787  df-res 4789

This theorem is referenced by:  rnco2  5089