Theorem rncoeq 5965

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 5964	. 2 ⊢ (dom ◡𝐵 = ran ◡𝐴 → dom (◡𝐵 ∘ ◡𝐴) = dom ◡𝐴)
2		eqcom 2767	. . 3 ⊢ (dom 𝐴 = ran 𝐵 ↔ ran 𝐵 = dom 𝐴)
3		df-rn 5666	. . . 4 ⊢ ran 𝐵 = dom ◡𝐵
4		dfdm4 5879	. . . 4 ⊢ dom 𝐴 = ran ◡𝐴
5	3, 4	eqeq12i 2778	. . 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 5666	. . . 4 ⊢ ran (𝐴 ∘ 𝐵) = dom ◡(𝐴 ∘ 𝐵)
8		cnvco 5869	. . . . 5 ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴)
9	8	dmeqi 5888	. . . 4 ⊢ dom ◡(𝐴 ∘ 𝐵) = dom (◡𝐵 ∘ ◡𝐴)
10	7, 9	eqtri 2783	. . 3 ⊢ ran (𝐴 ∘ 𝐵) = dom (◡𝐵 ∘ ◡𝐴)
11		df-rn 5666	. . 3 ⊢ ran 𝐴 = dom ◡𝐴
12	10, 11	eqeq12i 2778	. 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)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  ^◡ccnv 5654  dom cdm 5655  ran crn 5656   ∘ ccom 5659

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666

This theorem is used by:  dfdm2  6279  esplysply  34081  algextdeglem4  34230