Theorem imaeq12d 5065

Description: Equality theorem for image. (Contributed by Mario Carneiro, 4-Dec-2016.)

Hypotheses

Ref	Expression
imaeq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)
imaeq12d.2	⊢ (𝜑 → 𝐶 = 𝐷)

Assertion

Ref	Expression
imaeq12d	⊢ (𝜑 → (𝐴 “ 𝐶) = (𝐵 “ 𝐷))

Proof of Theorem imaeq12d

Step	Hyp	Ref	Expression
1		imaeq1d.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
2	1	imaeq1d 5063	. 2 ⊢ (𝜑 → (𝐴 “ 𝐶) = (𝐵 “ 𝐶))
3		imaeq12d.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
4	3	imaeq2d 5064	. 2 ⊢ (𝜑 → (𝐵 “ 𝐶) = (𝐵 “ 𝐷))
5	2, 4	eqtrd 2262	1 ⊢ (𝜑 → (𝐴 “ 𝐶) = (𝐵 “ 𝐷))

Syntax hints:   → wi 4   = wceq 1395   “ cima 4719

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-xp 4722  df-cnv 4724  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729

This theorem is referenced by:  csbima12g  5085  caseinl  7246  caseinr  7247  isunitd  14055