Theorem rncoeq 4605

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 4604	. 2 ⊢ (dom ◡𝐵 = ran ◡𝐴 → dom (◡𝐵 ∘ ◡𝐴) = dom ◡𝐴)
2		eqcom 2042	. . 3 ⊢ (dom 𝐴 = ran 𝐵 ↔ ran 𝐵 = dom 𝐴)
3		df-rn 4356	. . . 4 ⊢ ran 𝐵 = dom ◡𝐵
4		dfdm4 4527	. . . 4 ⊢ dom 𝐴 = ran ◡𝐴
5	3, 4	eqeq12i 2053	. . 3 ⊢ (ran 𝐵 = dom 𝐴 ↔ dom ◡𝐵 = ran ◡𝐴)
6	2, 5	bitri 173	. 2 ⊢ (dom 𝐴 = ran 𝐵 ↔ dom ◡𝐵 = ran ◡𝐴)
7		df-rn 4356	. . . 4 ⊢ ran (𝐴 ∘ 𝐵) = dom ◡(𝐴 ∘ 𝐵)
8		cnvco 4520	. . . . 5 ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴)
9	8	dmeqi 4536	. . . 4 ⊢ dom ◡(𝐴 ∘ 𝐵) = dom (◡𝐵 ∘ ◡𝐴)
10	7, 9	eqtri 2060	. . 3 ⊢ ran (𝐴 ∘ 𝐵) = dom (◡𝐵 ∘ ◡𝐴)
11		df-rn 4356	. . 3 ⊢ ran 𝐴 = dom ◡𝐴
12	10, 11	eqeq12i 2053	. 2 ⊢ (ran (𝐴 ∘ 𝐵) = ran 𝐴 ↔ dom (◡𝐵 ∘ ◡𝐴) = dom ◡𝐴)
13	1, 6, 12	3imtr4i 190	1 ⊢ (dom 𝐴 = ran 𝐵 → ran (𝐴 ∘ 𝐵) = ran 𝐴)

Syntax hints:   → wi 4   = wceq 1243  ^◡ccnv 4344  dom cdm 4345  ran crn 4346   ∘ ccom 4349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356

This theorem is referenced by:  dfdm2  4852  foco  5116