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Theorem xrneq1 36070
 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 5699 . . 3 (𝐴 = 𝐵 → ((1st ↾ (V × V)) ∘ 𝐴) = ((1st ↾ (V × V)) ∘ 𝐵))
21ineq1d 4117 . 2 (𝐴 = 𝐵 → (((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐶)) = (((1st ↾ (V × V)) ∘ 𝐵) ∩ ((2nd ↾ (V × V)) ∘ 𝐶)))
3 df-xrn 36064 . 2 (𝐴𝐶) = (((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐶))
4 df-xrn 36064 . 2 (𝐵𝐶) = (((1st ↾ (V × V)) ∘ 𝐵) ∩ ((2nd ↾ (V × V)) ∘ 𝐶))
52, 3, 43eqtr4g 2819 1 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1539  Vcvv 3410   ∩ cin 3858   × cxp 5523  ◡ccnv 5524   ↾ cres 5527   ∘ ccom 5529  1st c1st 7692  2nd c2nd 7693   ⋉ cxrn 35893 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1542  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-rab 3080  df-v 3412  df-in 3866  df-ss 3876  df-br 5034  df-opab 5096  df-co 5534  df-xrn 36064 This theorem is referenced by:  xrneq1i  36071  xrneq1d  36072  xrneq12  36076
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