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Theorem xrneq1 36771
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 5812 . . 3 (𝐴 = 𝐵 → ((1st ↾ (V × V)) ∘ 𝐴) = ((1st ↾ (V × V)) ∘ 𝐵))
21ineq1d 4169 . 2 (𝐴 = 𝐵 → (((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐶)) = (((1st ↾ (V × V)) ∘ 𝐵) ∩ ((2nd ↾ (V × V)) ∘ 𝐶)))
3 df-xrn 36765 . 2 (𝐴𝐶) = (((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐶))
4 df-xrn 36765 . 2 (𝐵𝐶) = (((1st ↾ (V × V)) ∘ 𝐵) ∩ ((2nd ↾ (V × V)) ∘ 𝐶))
52, 3, 43eqtr4g 2801 1 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  Vcvv 3443  cin 3907   × cxp 5629  ccnv 5630  cres 5633  ccom 5635  1st c1st 7911  2nd c2nd 7912  cxrn 36565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3406  df-v 3445  df-in 3915  df-ss 3925  df-br 5104  df-opab 5166  df-co 5640  df-xrn 36765
This theorem is referenced by:  xrneq1i  36772  xrneq1d  36773  xrneq12  36777
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