Theorem rnco 6199

Description: The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) Avoid ax-11 2160. (Revised by TM, 24-Jan-2026.)

Assertion

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

Proof of Theorem rnco

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

Step	Hyp	Ref	Expression
1		vex 3440	. . . . . 6 ⊢ 𝑥 ∈ V
2		vex 3440	. . . . . 6 ⊢ 𝑦 ∈ V
3	1, 2	brco 5810	. . . . 5 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))
4	3	exbii 1849	. . . 4 ⊢ (∃𝑥 𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑥∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))
5		breq1 5094	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑥𝐵𝑧 ↔ 𝑤𝐵𝑧))
6	5	anbi1d 631	. . . . 5 ⊢ (𝑥 = 𝑤 → ((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ (𝑤𝐵𝑧 ∧ 𝑧𝐴𝑦)))
7		breq2 5095	. . . . . 6 ⊢ (𝑧 = 𝑤 → (𝑥𝐵𝑧 ↔ 𝑥𝐵𝑤))
8		breq1 5094	. . . . . 6 ⊢ (𝑧 = 𝑤 → (𝑧𝐴𝑦 ↔ 𝑤𝐴𝑦))
9	7, 8	anbi12d 632	. . . . 5 ⊢ (𝑧 = 𝑤 → ((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ (𝑥𝐵𝑤 ∧ 𝑤𝐴𝑦)))
10	6, 9	excomw 2047	. . . 4 ⊢ (∃𝑥∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧∃𝑥(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))
11		vex 3440	. . . . . . . 8 ⊢ 𝑧 ∈ V
12	11	elrn 5833	. . . . . . 7 ⊢ (𝑧 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝑧)
13	12	anbi1i 624	. . . . . 6 ⊢ ((𝑧 ∈ ran 𝐵 ∧ 𝑧𝐴𝑦) ↔ (∃𝑥 𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))
14	2	brresi 5937	. . . . . 6 ⊢ (𝑧(𝐴 ↾ ran 𝐵)𝑦 ↔ (𝑧 ∈ ran 𝐵 ∧ 𝑧𝐴𝑦))
15		19.41v 1950	. . . . . 6 ⊢ (∃𝑥(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ (∃𝑥 𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))
16	13, 14, 15	3bitr4ri 304	. . . . 5 ⊢ (∃𝑥(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ 𝑧(𝐴 ↾ ran 𝐵)𝑦)
17	16	exbii 1849	. . . 4 ⊢ (∃𝑧∃𝑥(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦)
18	4, 10, 17	3bitri 297	. . 3 ⊢ (∃𝑥 𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦)
19	2	elrn 5833	. . 3 ⊢ (𝑦 ∈ ran (𝐴 ∘ 𝐵) ↔ ∃𝑥 𝑥(𝐴 ∘ 𝐵)𝑦)
20	2	elrn 5833	. . 3 ⊢ (𝑦 ∈ ran (𝐴 ↾ ran 𝐵) ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦)
21	18, 19, 20	3bitr4i 303	. 2 ⊢ (𝑦 ∈ ran (𝐴 ∘ 𝐵) ↔ 𝑦 ∈ ran (𝐴 ↾ ran 𝐵))
22	21	eqriv 2728	1 ⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵)

Syntax hints:   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2111   class class class wbr 5091  ran crn 5617   ↾ cres 5618   ∘ ccom 5620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-xp 5622  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628

This theorem is referenced by:  rnco2  6201  coeq0  6203  focofo  6748  cofunexg  7881  1stcof  7951  2ndcof  7952  smobeth  10474  cycpmconjv  33106  elmsubrn  35560  ftc1anclem3  37734