Theorem imaeq1d 4610

Description: Equality theorem for image. (Contributed by FL, 15-Dec-2006.)

Hypothesis

Ref	Expression
imaeq1d.1	⊢ (φ → A = B)

Assertion

Ref	Expression
imaeq1d	⊢ (φ → (A “ 𝐶) = (B “ 𝐶))

Proof of Theorem imaeq1d

Step	Hyp	Ref	Expression
1		imaeq1d.1	. 2 ⊢ (φ → A = B)
2		imaeq1 4606	. 2 ⊢ (A = B → (A “ 𝐶) = (B “ 𝐶))
3	1, 2	syl 14	1 ⊢ (φ → (A “ 𝐶) = (B “ 𝐶))

Syntax hints:   → wi 4   = wceq 1242   “ cima 4291

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301

This theorem is referenced by:  imaeq12d  4612  nfimad  4620  f1imacnv  5086  foimacnv  5087  suppssof1  5670