Theorem rnco 6252

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 2194. (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 3457	. . . . . 6 ⊢ 𝑥 ∈ V
2		vex 3457	. . . . . 6 ⊢ 𝑦 ∈ V
3	1, 2	brco 5854	. . . . 5 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))
4	3	exbii 1881	. . . 4 ⊢ (∃𝑥 𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑥∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))
5		breq1 5110	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑥𝐵𝑧 ↔ 𝑤𝐵𝑧))
6	5	anbi1d 643	. . . . 5 ⊢ (𝑥 = 𝑤 → ((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ (𝑤𝐵𝑧 ∧ 𝑧𝐴𝑦)))
7		breq2 5111	. . . . . 6 ⊢ (𝑧 = 𝑤 → (𝑥𝐵𝑧 ↔ 𝑥𝐵𝑤))
8		breq1 5110	. . . . . 6 ⊢ (𝑧 = 𝑤 → (𝑧𝐴𝑦 ↔ 𝑤𝐴𝑦))
9	7, 8	anbi12d 644	. . . . 5 ⊢ (𝑧 = 𝑤 → ((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ (𝑥𝐵𝑤 ∧ 𝑤𝐴𝑦)))
10	6, 9	excomw 2079	. . . 4 ⊢ (∃𝑥∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧∃𝑥(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))
11		vex 3457	. . . . . . . 8 ⊢ 𝑧 ∈ V
12	11	elrn 5881	. . . . . . 7 ⊢ (𝑧 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝑧)
13	12	anbi1i 636	. . . . . 6 ⊢ ((𝑧 ∈ ran 𝐵 ∧ 𝑧𝐴𝑦) ↔ (∃𝑥 𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))
14	2	brresi 5985	. . . . . 6 ⊢ (𝑧(𝐴 ↾ ran 𝐵)𝑦 ↔ (𝑧 ∈ ran 𝐵 ∧ 𝑧𝐴𝑦))
15		19.41v 1982	. . . . . 6 ⊢ (∃𝑥(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ (∃𝑥 𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))
16	13, 14, 15	3bitr4ri 307	. . . . 5 ⊢ (∃𝑥(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ 𝑧(𝐴 ↾ ran 𝐵)𝑦)
17	16	exbii 1881	. . . 4 ⊢ (∃𝑧∃𝑥(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦)
18	4, 10, 17	3bitri 300	. . 3 ⊢ (∃𝑥 𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦)
19	2	elrn 5881	. . 3 ⊢ (𝑦 ∈ ran (𝐴 ∘ 𝐵) ↔ ∃𝑥 𝑥(𝐴 ∘ 𝐵)𝑦)
20	2	elrn 5881	. . 3 ⊢ (𝑦 ∈ ran (𝐴 ↾ ran 𝐵) ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦)
21	18, 19, 20	3bitr4i 306	. 2 ⊢ (𝑦 ∈ ran (𝐴 ∘ 𝐵) ↔ 𝑦 ∈ ran (𝐴 ↾ ran 𝐵))
22	21	eqriv 2759	1 ⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145   class class class wbr 5107  ran crn 5660   ↾ cres 5661   ∘ ccom 5663

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-xp 5665  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671

This theorem is used by:  rnco2  6254  coeq0  6256  focofo  6806  cofunexg  7950  1stcof  8020  2ndcof  8021  smobeth  10599  cycpmconjv  33590  elmsubrn  36115  ftc1anclem3  38452