Theorem imaeq12d 4882

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 4880	. 2 ⊢ (𝜑 → (𝐴 “ 𝐶) = (𝐵 “ 𝐶))
3		imaeq12d.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
4	3	imaeq2d 4881	. 2 ⊢ (𝜑 → (𝐵 “ 𝐶) = (𝐵 “ 𝐷))
5	2, 4	eqtrd 2172	1 ⊢ (𝜑 → (𝐴 “ 𝐶) = (𝐵 “ 𝐷))

Syntax hints:   → wi 4   = wceq 1331   “ cima 4542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-cnv 4547  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552

This theorem is referenced by:  csbima12g  4900  caseinl  6976  caseinr  6977