Theorem imaeq12d 6063

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 6061	. 2 ⊢ (𝜑 → (𝐴 “ 𝐶) = (𝐵 “ 𝐶))
3		imaeq12d.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
4	3	imaeq2d 6062	. 2 ⊢ (𝜑 → (𝐵 “ 𝐶) = (𝐵 “ 𝐷))
5	2, 4	eqtrd 2798	1 ⊢ (𝜑 → (𝐴 “ 𝐶) = (𝐵 “ 𝐷))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   “ cima 5664

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-xp 5667  df-cnv 5669  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674

This theorem is used by:  csbima12  6081  predeq123  6303  vdwpc  17044  dmdprd  20074  isunit  20460  qtopval  23861  limciun  26062  ig1pval  26342  ispth  30079  esplyval  33961  irngval  34084  qqhval  34371  eulerpartgbij  34771  orvcval  34857  ballotlemrval  34917  ballotlemrinv0  34932  ballotlemrinv  34933  mthmval  36075  bj-projeq  37656  itg2addnclem2  38351  islmodfg  43824  heeq12  44530  isgrim  48675  imaf1hom  49914  imaidfu  49916  imasubc  49957  imassc  49959  imaid  49960