Theorem imaeq12d 6055

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 6053	. 2 ⊢ (𝜑 → (𝐴 “ 𝐶) = (𝐵 “ 𝐶))
3		imaeq12d.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
4	3	imaeq2d 6054	. 2 ⊢ (𝜑 → (𝐵 “ 𝐶) = (𝐵 “ 𝐷))
5	2, 4	eqtrd 2796	1 ⊢ (𝜑 → (𝐴 “ 𝐶) = (𝐵 “ 𝐷))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   “ cima 5654

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664

This theorem is used by:  csbima12  6073  predeq123  6298  vdwpc  17138  dmdprd  20194  isunit  20583  qtopval  23994  limciun  26194  ig1pval  26474  ispth  30288  esplyval  34176  irngval  34299  qqhval  34586  eulerpartgbij  34987  orvcval  35073  ballotlemrval  35133  ballotlemrinv0  35148  ballotlemrinv  35149  mthmval  36309  bj-projeq  37875  itg2addnclem2  38558  islmodfg  44029  heeq12  44735  isgrim  48924  imaf1hom  50160  imaidfu  50162  imasubc  50203  imassc  50205  imaid  50206