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Theorem xrneq2 35512
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 5722 . . 3 (𝐴 = 𝐵 → ((2nd ↾ (V × V)) ∘ 𝐴) = ((2nd ↾ (V × V)) ∘ 𝐵))
21ineq2d 4186 . 2 (𝐴 = 𝐵 → (((1st ↾ (V × V)) ∘ 𝐶) ∩ ((2nd ↾ (V × V)) ∘ 𝐴)) = (((1st ↾ (V × V)) ∘ 𝐶) ∩ ((2nd ↾ (V × V)) ∘ 𝐵)))
3 df-xrn 35503 . 2 (𝐶𝐴) = (((1st ↾ (V × V)) ∘ 𝐶) ∩ ((2nd ↾ (V × V)) ∘ 𝐴))
4 df-xrn 35503 . 2 (𝐶𝐵) = (((1st ↾ (V × V)) ∘ 𝐶) ∩ ((2nd ↾ (V × V)) ∘ 𝐵))
52, 3, 43eqtr4g 2878 1 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  Vcvv 3492  cin 3932   × cxp 5546  ccnv 5547  cres 5550  ccom 5552  1st c1st 7676  2nd c2nd 7677  cxrn 35333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-in 3940  df-ss 3949  df-br 5058  df-opab 5120  df-co 5557  df-xrn 35503
This theorem is referenced by:  xrneq2i  35513  xrneq2d  35514  xrneq12  35515
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