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Theorem xrneq1 34482
 Description: Equality theorem for the range Cartesian product. (Contributed by Peter Mazsa, 16-Dec-2020.)
Assertion
Ref Expression
xrneq1 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))

Proof of Theorem xrneq1
StepHypRef Expression
1 coeq2 5420 . . 3 (𝐴 = 𝐵 → ((1st ↾ (V × V)) ∘ 𝐴) = ((1st ↾ (V × V)) ∘ 𝐵))
21ineq1d 3965 . 2 (𝐴 = 𝐵 → (((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐶)) = (((1st ↾ (V × V)) ∘ 𝐵) ∩ ((2nd ↾ (V × V)) ∘ 𝐶)))
3 df-xrn 34476 . 2 (𝐴𝐶) = (((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐶))
4 df-xrn 34476 . 2 (𝐵𝐶) = (((1st ↾ (V × V)) ∘ 𝐵) ∩ ((2nd ↾ (V × V)) ∘ 𝐶))
52, 3, 43eqtr4g 2830 1 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1631  Vcvv 3351   ∩ cin 3723   × cxp 5248  ◡ccnv 5249   ↾ cres 5252   ∘ ccom 5254  1st c1st 7314  2nd c2nd 7315   ⋉ cxrn 34315 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-in 3731  df-ss 3738  df-br 4788  df-opab 4848  df-co 5259  df-xrn 34476 This theorem is referenced by:  xrneq1i  34483  xrneq1d  34484  xrneq12  34488
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