Theorem rncoeq 5939

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 5938	. 2 ⊢ (dom ◡𝐵 = ran ◡𝐴 → dom (◡𝐵 ∘ ◡𝐴) = dom ◡𝐴)
2		eqcom 2744	. . 3 ⊢ (dom 𝐴 = ran 𝐵 ↔ ran 𝐵 = dom 𝐴)
3		df-rn 5643	. . . 4 ⊢ ran 𝐵 = dom ◡𝐵
4		dfdm4 5852	. . . 4 ⊢ dom 𝐴 = ran ◡𝐴
5	3, 4	eqeq12i 2755	. . 3 ⊢ (ran 𝐵 = dom 𝐴 ↔ dom ◡𝐵 = ran ◡𝐴)
6	2, 5	bitri 275	. 2 ⊢ (dom 𝐴 = ran 𝐵 ↔ dom ◡𝐵 = ran ◡𝐴)
7		df-rn 5643	. . . 4 ⊢ ran (𝐴 ∘ 𝐵) = dom ◡(𝐴 ∘ 𝐵)
8		cnvco 5842	. . . . 5 ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴)
9	8	dmeqi 5861	. . . 4 ⊢ dom ◡(𝐴 ∘ 𝐵) = dom (◡𝐵 ∘ ◡𝐴)
10	7, 9	eqtri 2760	. . 3 ⊢ ran (𝐴 ∘ 𝐵) = dom (◡𝐵 ∘ ◡𝐴)
11		df-rn 5643	. . 3 ⊢ ran 𝐴 = dom ◡𝐴
12	10, 11	eqeq12i 2755	. 2 ⊢ (ran (𝐴 ∘ 𝐵) = ran 𝐴 ↔ dom (◡𝐵 ∘ ◡𝐴) = dom ◡𝐴)
13	1, 6, 12	3imtr4i 292	1 ⊢ (dom 𝐴 = ran 𝐵 → ran (𝐴 ∘ 𝐵) = ran 𝐴)

Syntax hints:   → wi 4   = wceq 1542  ^◡ccnv 5631  dom cdm 5632  ran crn 5633   ∘ ccom 5636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643

This theorem is referenced by:  dfdm2  6247  esplysply  33748  algextdeglem4  33898