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Theorem xrneq1 38378
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 5869 . . 3 (𝐴 = 𝐵 → ((1st ↾ (V × V)) ∘ 𝐴) = ((1st ↾ (V × V)) ∘ 𝐵))
21ineq1d 4219 . 2 (𝐴 = 𝐵 → (((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐶)) = (((1st ↾ (V × V)) ∘ 𝐵) ∩ ((2nd ↾ (V × V)) ∘ 𝐶)))
3 df-xrn 38372 . 2 (𝐴𝐶) = (((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐶))
4 df-xrn 38372 . 2 (𝐵𝐶) = (((1st ↾ (V × V)) ∘ 𝐵) ∩ ((2nd ↾ (V × V)) ∘ 𝐶))
52, 3, 43eqtr4g 2802 1 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  Vcvv 3480  cin 3950   × cxp 5683  ccnv 5684  cres 5687  ccom 5689  1st c1st 8012  2nd c2nd 8013  cxrn 38181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-in 3958  df-ss 3968  df-br 5144  df-opab 5206  df-co 5694  df-xrn 38372
This theorem is referenced by:  xrneq1i  38379  xrneq1d  38380  xrneq12  38384
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