Theorem rncoeq 2574

Description: Range of a composition.

Assertion

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

Proof of Theorem rncoeq

Step	Hyp	Ref	Expression
1		dmcoeq 2573	. 2 ⊢ (dom ◡B = ran ◡A → dom (◡B ∘ ◡A) = dom ◡A)
2		cleqcom 1103	. . 3 ⊢ (dom A = ran B ↔ ran B = dom A)
3		df-rn 2429	. . . 4 ⊢ ran B = dom ◡B
4		dfdm4 2525	. . . 4 ⊢ dom A = ran ◡A
5	3, 4	cleq12i 1114	. . 3 ⊢ (ran B = dom A ↔ dom ◡B = ran ◡A)
6	2, 5	bitr 151	. 2 ⊢ (dom A = ran B ↔ dom ◡B = ran ◡A)
7		df-rn 2429	. . . 4 ⊢ ran (A ∘ B) = dom ◡(A ∘ B)
8		cnvco 2520	. . . . 5 ⊢ ◡(A ∘ B) = (◡B ∘ ◡A)
9	8	dmeqi 2532	. . . 4 ⊢ dom ◡(A ∘ B) = dom (◡B ∘ ◡A)
10	7, 9	eqtr 1119	. . 3 ⊢ ran (A ∘ B) = dom (◡B ∘ ◡A)
11		df-rn 2429	. . 3 ⊢ ran A = dom ◡A
12	10, 11	cleq12i 1114	. 2 ⊢ (ran (A ∘ B) = ran A ↔ dom (◡B ∘ ◡A) = dom ◡A)
13	1, 6, 12	3imtr4 192	1 ⊢ (dom A = ran B → ran (A ∘ B) = ran A)

Syntax hints:   → wi 2   = wceq 1091  ^◡ccnv 2409  dom cdm 2410  ran crn 2411   ∘ ccom 2414

This theorem is referenced by:  f1oco 2816

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429

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