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Theorem xrneq1 36652
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 5800 . . 3 (𝐴 = 𝐵 → ((1st ↾ (V × V)) ∘ 𝐴) = ((1st ↾ (V × V)) ∘ 𝐵))
21ineq1d 4158 . 2 (𝐴 = 𝐵 → (((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐶)) = (((1st ↾ (V × V)) ∘ 𝐵) ∩ ((2nd ↾ (V × V)) ∘ 𝐶)))
3 df-xrn 36646 . 2 (𝐴𝐶) = (((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐶))
4 df-xrn 36646 . 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 1540  Vcvv 3441  cin 3897   × cxp 5618  ccnv 5619  cres 5622  ccom 5624  1st c1st 7897  2nd c2nd 7898  cxrn 36445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3404  df-v 3443  df-in 3905  df-ss 3915  df-br 5093  df-opab 5155  df-co 5629  df-xrn 36646
This theorem is referenced by:  xrneq1i  36653  xrneq1d  36654  xrneq12  36658
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