Theorem rncoeq 5829

Description: Range of a composition. (Contributed by NM, 19-Mar-1998.)

Assertion

Ref	Expression
rncoeq	⊢ (dom 𝐴 = ran 𝐵 → ran (𝐴 ∘ 𝐵) = ran 𝐴)

Proof of Theorem rncoeq

Step	Hyp	Ref	Expression
1		dmcoeq 5828	. 2 ⊢ (dom ◡𝐵 = ran ◡𝐴 → dom (◡𝐵 ∘ ◡𝐴) = dom ◡𝐴)
2		eqcom 2743	. . 3 ⊢ (dom 𝐴 = ran 𝐵 ↔ ran 𝐵 = dom 𝐴)
3		df-rn 5547	. . . 4 ⊢ ran 𝐵 = dom ◡𝐵
4		dfdm4 5749	. . . 4 ⊢ dom 𝐴 = ran ◡𝐴
5	3, 4	eqeq12i 2751	. . 3 ⊢ (ran 𝐵 = dom 𝐴 ↔ dom ◡𝐵 = ran ◡𝐴)
6	2, 5	bitri 278	. 2 ⊢ (dom 𝐴 = ran 𝐵 ↔ dom ◡𝐵 = ran ◡𝐴)
7		df-rn 5547	. . . 4 ⊢ ran (𝐴 ∘ 𝐵) = dom ◡(𝐴 ∘ 𝐵)
8		cnvco 5739	. . . . 5 ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴)
9	8	dmeqi 5758	. . . 4 ⊢ dom ◡(𝐴 ∘ 𝐵) = dom (◡𝐵 ∘ ◡𝐴)
10	7, 9	eqtri 2759	. . 3 ⊢ ran (𝐴 ∘ 𝐵) = dom (◡𝐵 ∘ ◡𝐴)
11		df-rn 5547	. . 3 ⊢ ran 𝐴 = dom ◡𝐴
12	10, 11	eqeq12i 2751	. 2 ⊢ (ran (𝐴 ∘ 𝐵) = ran 𝐴 ↔ dom (◡𝐵 ∘ ◡𝐴) = dom ◡𝐴)
13	1, 6, 12	3imtr4i 295	1 ⊢ (dom 𝐴 = ran 𝐵 → ran (𝐴 ∘ 𝐵) = ran 𝐴)

Syntax hints:   → wi 4   = wceq 1543  ^◡ccnv 5535  dom cdm 5536  ran crn 5537   ∘ ccom 5540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-br 5040  df-opab 5102  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547

This theorem is referenced by:  dfdm2  6124