Theorem rncoeq 3367

Description: Range of a composition.

Assertion

Ref	Expression
rncoeq	|- (dom A = ran B -> ran ( A o. B) = ran A)

Proof of Theorem rncoeq

Step	Hyp	Ref	Expression
1		dmcoeq 3366	. 2 |- (dom `' B = ran `' A -> dom (`'B o. `'A) = dom `' A)
2		eqcom 1477	. . 3 |- (dom A = ran B <-> ran B = dom A)
3		df-rn 3189	. . . 4 |- ran B = dom `' B
4		dfdm4 3305	. . . 4 |- dom A = ran `' A
5	3, 4	eqeq12i 1488	. . 3 |- (ran B = dom A <-> dom `' B = ran `' A)
6	2, 5	bitr 173	. 2 |- (dom A = ran B <-> dom `' B = ran `' A)
7		df-rn 3189	. . . 4 |- ran ( A o. B) = dom `'(A o. B)
8		cnvco 3300	. . . . 5 |- `'(A o. B) = (`'B o. `'A)
9	8	dmeqi 3312	. . . 4 |- dom `'(A o. B) = dom (`'B o. `'A)
10	7, 9	eqtr 1495	. . 3 |- ran ( A o. B) = dom (`'B o. `'A)
11		df-rn 3189	. . 3 |- ran A = dom `' A
12	10, 11	eqeq12i 1488	. 2 |- (ran ( A o. B) = ran A <-> dom (`'B o. `'A) = dom `' A)
13	1, 6, 12	3imtr4 219	1 |- (dom A = ran B -> ran ( A o. B) = ran A)

Syntax hints:   -> wi 3   = wceq 956  `'ccnv 3169  dom cdm 3170  ran crn 3171   o. ccom 3174

This theorem is referenced by:  foco 3682

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189

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