Theorem rnco 6170

Description: The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)

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 3441	. . . . . 6 ⊢ 𝑥 ∈ V
2		vex 3441	. . . . . 6 ⊢ 𝑦 ∈ V
3	1, 2	brco 5792	. . . . 5 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))
4	3	exbii 1848	. . . 4 ⊢ (∃𝑥 𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑥∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))
5		excom 2160	. . . 4 ⊢ (∃𝑥∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧∃𝑥(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))
6		vex 3441	. . . . . . . 8 ⊢ 𝑧 ∈ V
7	6	elrn 5815	. . . . . . 7 ⊢ (𝑧 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝑧)
8	7	anbi1i 625	. . . . . 6 ⊢ ((𝑧 ∈ ran 𝐵 ∧ 𝑧𝐴𝑦) ↔ (∃𝑥 𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))
9	2	brresi 5912	. . . . . 6 ⊢ (𝑧(𝐴 ↾ ran 𝐵)𝑦 ↔ (𝑧 ∈ ran 𝐵 ∧ 𝑧𝐴𝑦))
10		19.41v 1951	. . . . . 6 ⊢ (∃𝑥(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ (∃𝑥 𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))
11	8, 9, 10	3bitr4ri 304	. . . . 5 ⊢ (∃𝑥(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ 𝑧(𝐴 ↾ ran 𝐵)𝑦)
12	11	exbii 1848	. . . 4 ⊢ (∃𝑧∃𝑥(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦)
13	4, 5, 12	3bitri 297	. . 3 ⊢ (∃𝑥 𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦)
14	2	elrn 5815	. . 3 ⊢ (𝑦 ∈ ran (𝐴 ∘ 𝐵) ↔ ∃𝑥 𝑥(𝐴 ∘ 𝐵)𝑦)
15	2	elrn 5815	. . 3 ⊢ (𝑦 ∈ ran (𝐴 ↾ ran 𝐵) ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦)
16	13, 14, 15	3bitr4i 303	. 2 ⊢ (𝑦 ∈ ran (𝐴 ∘ 𝐵) ↔ 𝑦 ∈ ran (𝐴 ↾ ran 𝐵))
17	16	eqriv 2733	1 ⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵)

Syntax hints:   ∧ wa 397   = wceq 1539  ∃wex 1779   ∈ wcel 2104   class class class wbr 5081  ran crn 5601   ↾ cres 5602   ∘ ccom 5604

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-11 2152  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3063  df-rex 3072  df-rab 3306  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-br 5082  df-opab 5144  df-xp 5606  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612

This theorem is referenced by:  rnco2  6171  coeq0  6173  focofo  6731  cofunexg  7823  1stcof  7893  2ndcof  7894  smobeth  10392  cycpmconjv  31458  elmsubrn  33539  ftc1anclem3  35900