Description: Equality theorem for range. (Contributed by NM, 29-Dec-1996)
Ref | Expression | ||
---|---|---|---|
Assertion | rneq | ⊢ ( 𝐴 = 𝐵 → ran 𝐴 = ran 𝐵 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnveq | ⊢ ( 𝐴 = 𝐵 → ^{◡} 𝐴 = ^{◡} 𝐵 ) | |
2 | 1 | dmeqd | ⊢ ( 𝐴 = 𝐵 → dom ^{◡} 𝐴 = dom ^{◡} 𝐵 ) |
3 | df-rn | ⊢ ran 𝐴 = dom ^{◡} 𝐴 | |
4 | df-rn | ⊢ ran 𝐵 = dom ^{◡} 𝐵 | |
5 | 2 3 4 | 3eqtr4g | ⊢ ( 𝐴 = 𝐵 → ran 𝐴 = ran 𝐵 ) |