Theorem imaeq12d 6054

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 6052	. 2 ⊢ (𝜑 → (𝐴 “ 𝐶) = (𝐵 “ 𝐶))
3		imaeq12d.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
4	3	imaeq2d 6053	. 2 ⊢ (𝜑 → (𝐵 “ 𝐶) = (𝐵 “ 𝐷))
5	2, 4	eqtrd 2800	1 ⊢ (𝜑 → (𝐴 “ 𝐶) = (𝐵 “ 𝐷))

Syntax hints:   → wi 4   = wceq 1563   “ cima 5655

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-xp 5658  df-cnv 5660  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665

This theorem is referenced by:  csbima12  6072  predeq123  6293  vdwpc  17030  dmdprd  20061  isunit  20446  qtopval  23813  limciun  26014  ig1pval  26294  ispth  29979  esplyval  33869  irngval  33992  qqhval  34279  eulerpartgbij  34679  orvcval  34765  ballotlemrval  34825  ballotlemrinv0  34840  ballotlemrinv  34841  mthmval  35938  bj-projeq  37489  itg2addnclem2  38183  islmodfg  43658  heeq12  44364  isgrim  48502  imaf1hom  49737  imaidfu  49739  imasubc  49780  imassc  49782  imaid  49783