Theorem rnco 5235

Description: The range of the composition of two classes. (Contributed by NM, 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		vex 2802	. . . . . 6 ⊢ 𝑥 ∈ V
2		vex 2802	. . . . . 6 ⊢ 𝑦 ∈ V
3	1, 2	brco 4893	. . . . 5 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))
4	3	exbii 1651	. . . 4 ⊢ (∃𝑥 𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑥∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))
5		excom 1710	. . . 4 ⊢ (∃𝑥∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧∃𝑥(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))
6		ancom 266	. . . . . . 7 ⊢ ((∃𝑥 𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ (𝑧𝐴𝑦 ∧ ∃𝑥 𝑥𝐵𝑧))
7		19.41v 1949	. . . . . . 7 ⊢ (∃𝑥(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ (∃𝑥 𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))
8		vex 2802	. . . . . . . . 9 ⊢ 𝑧 ∈ V
9	8	elrn 4967	. . . . . . . 8 ⊢ (𝑧 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝑧)
10	9	anbi2i 457	. . . . . . 7 ⊢ ((𝑧𝐴𝑦 ∧ 𝑧 ∈ ran 𝐵) ↔ (𝑧𝐴𝑦 ∧ ∃𝑥 𝑥𝐵𝑧))
11	6, 7, 10	3bitr4i 212	. . . . . 6 ⊢ (∃𝑥(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ (𝑧𝐴𝑦 ∧ 𝑧 ∈ ran 𝐵))
12	2	brres 5011	. . . . . 6 ⊢ (𝑧(𝐴 ↾ ran 𝐵)𝑦 ↔ (𝑧𝐴𝑦 ∧ 𝑧 ∈ ran 𝐵))
13	11, 12	bitr4i 187	. . . . 5 ⊢ (∃𝑥(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ 𝑧(𝐴 ↾ ran 𝐵)𝑦)
14	13	exbii 1651	. . . 4 ⊢ (∃𝑧∃𝑥(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦)
15	4, 5, 14	3bitri 206	. . 3 ⊢ (∃𝑥 𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦)
16	2	elrn 4967	. . 3 ⊢ (𝑦 ∈ ran (𝐴 ∘ 𝐵) ↔ ∃𝑥 𝑥(𝐴 ∘ 𝐵)𝑦)
17	2	elrn 4967	. . 3 ⊢ (𝑦 ∈ ran (𝐴 ↾ ran 𝐵) ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦)
18	15, 16, 17	3bitr4i 212	. 2 ⊢ (𝑦 ∈ ran (𝐴 ∘ 𝐵) ↔ 𝑦 ∈ ran (𝐴 ↾ ran 𝐵))
19	18	eqriv 2226	1 ⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵)

Syntax hints:   ∧ wa 104   = wceq 1395  ∃wex 1538   ∈ wcel 2200   class class class wbr 4083  ran crn 4720   ↾ cres 4721   ∘ ccom 4723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-xp 4725  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731

This theorem is referenced by:  rnco2  5236  cofunexg  6260  1stcof  6315  2ndcof  6316  djudom  7268