![]() |
Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > xrneq1 | Structured version Visualization version GIF version |
Description: Equality theorem for the range Cartesian product. (Contributed by Peter Mazsa, 16-Dec-2020.) |
Ref | Expression |
---|---|
xrneq1 | ⊢ (𝐴 = 𝐵 → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coeq2 5872 | . . 3 ⊢ (𝐴 = 𝐵 → (◡(1st ↾ (V × V)) ∘ 𝐴) = (◡(1st ↾ (V × V)) ∘ 𝐵)) | |
2 | 1 | ineq1d 4227 | . 2 ⊢ (𝐴 = 𝐵 → ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐶)) = ((◡(1st ↾ (V × V)) ∘ 𝐵) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐶))) |
3 | df-xrn 38353 | . 2 ⊢ (𝐴 ⋉ 𝐶) = ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐶)) | |
4 | df-xrn 38353 | . 2 ⊢ (𝐵 ⋉ 𝐶) = ((◡(1st ↾ (V × V)) ∘ 𝐵) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐶)) | |
5 | 2, 3, 4 | 3eqtr4g 2800 | 1 ⊢ (𝐴 = 𝐵 → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 Vcvv 3478 ∩ cin 3962 × cxp 5687 ◡ccnv 5688 ↾ cres 5691 ∘ ccom 5693 1st c1st 8011 2nd c2nd 8012 ⋉ cxrn 38161 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-in 3970 df-ss 3980 df-br 5149 df-opab 5211 df-co 5698 df-xrn 38353 |
This theorem is referenced by: xrneq1i 38360 xrneq1d 38361 xrneq12 38365 |
Copyright terms: Public domain | W3C validator |