Theorem rnco 6243

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 3461	. . . . . 6 ⊢ 𝑥 ∈ V
2		vex 3461	. . . . . 6 ⊢ 𝑦 ∈ V
3	1, 2	brco 5847	. . . . 5 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))
4	3	exbii 1871	. . . 4 ⊢ (∃𝑥 𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑥∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))
5		breq1 5108	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑥𝐵𝑧 ↔ 𝑤𝐵𝑧))
6	5	anbi1d 642	. . . . 5 ⊢ (𝑥 = 𝑤 → ((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ (𝑤𝐵𝑧 ∧ 𝑧𝐴𝑦)))
7		breq2 5109	. . . . . 6 ⊢ (𝑧 = 𝑤 → (𝑥𝐵𝑧 ↔ 𝑥𝐵𝑤))
8		breq1 5108	. . . . . 6 ⊢ (𝑧 = 𝑤 → (𝑧𝐴𝑦 ↔ 𝑤𝐴𝑦))
9	7, 8	anbi12d 643	. . . . 5 ⊢ (𝑧 = 𝑤 → ((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ (𝑥𝐵𝑤 ∧ 𝑤𝐴𝑦)))
10	6, 9	excomw 2069	. . . 4 ⊢ (∃𝑥∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧∃𝑥(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))
11		vex 3461	. . . . . . . 8 ⊢ 𝑧 ∈ V
12	11	elrn 5874	. . . . . . 7 ⊢ (𝑧 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝑧)
13	12	anbi1i 635	. . . . . 6 ⊢ ((𝑧 ∈ ran 𝐵 ∧ 𝑧𝐴𝑦) ↔ (∃𝑥 𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))
14	2	brresi 5978	. . . . . 6 ⊢ (𝑧(𝐴 ↾ ran 𝐵)𝑦 ↔ (𝑧 ∈ ran 𝐵 ∧ 𝑧𝐴𝑦))
15		19.41v 1972	. . . . . 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 1871	. . . 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 5874	. . 3 ⊢ (𝑦 ∈ ran (𝐴 ∘ 𝐵) ↔ ∃𝑥 𝑥(𝐴 ∘ 𝐵)𝑦)
20	2	elrn 5874	. . 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 2762	1 ⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵)

Syntax hints:   ∧ wa 400   = wceq 1563  ∃wex 1802   ∈ wcel 2145   class class class wbr 5105  ran crn 5653   ↾ cres 5654   ∘ ccom 5656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-xp 5658  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664

This theorem is referenced by:  rnco2  6245  coeq0  6247  focofo  6795  cofunexg  7934  1stcof  8004  2ndcof  8005  smobeth  10559  cycpmconjv  33375  elmsubrn  35891  ftc1anclem3  38206