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Theorem xrneq2 38362
Description: Equality theorem for the range Cartesian product. (Contributed by Peter Mazsa, 16-Dec-2020.)
Assertion
Ref Expression
xrneq2 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))

Proof of Theorem xrneq2
StepHypRef Expression
1 coeq2 5872 . . 3 (𝐴 = 𝐵 → ((2nd ↾ (V × V)) ∘ 𝐴) = ((2nd ↾ (V × V)) ∘ 𝐵))
21ineq2d 4228 . 2 (𝐴 = 𝐵 → (((1st ↾ (V × V)) ∘ 𝐶) ∩ ((2nd ↾ (V × V)) ∘ 𝐴)) = (((1st ↾ (V × V)) ∘ 𝐶) ∩ ((2nd ↾ (V × V)) ∘ 𝐵)))
3 df-xrn 38353 . 2 (𝐶𝐴) = (((1st ↾ (V × V)) ∘ 𝐶) ∩ ((2nd ↾ (V × V)) ∘ 𝐴))
4 df-xrn 38353 . 2 (𝐶𝐵) = (((1st ↾ (V × V)) ∘ 𝐶) ∩ ((2nd ↾ (V × V)) ∘ 𝐵))
52, 3, 43eqtr4g 2800 1 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  Vcvv 3478  cin 3962   × cxp 5687  ccnv 5688  cres 5691  ccom 5693  1st c1st 8011  2nd c2nd 8012  cxrn 38161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-in 3970  df-ss 3980  df-br 5149  df-opab 5211  df-co 5698  df-xrn 38353
This theorem is referenced by:  xrneq2i  38363  xrneq2d  38364  xrneq12  38365
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