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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 5800 | . . 3 ⊢ (𝐴 = 𝐵 → (◡(1st ↾ (V × V)) ∘ 𝐴) = (◡(1st ↾ (V × V)) ∘ 𝐵)) | |
2 | 1 | ineq1d 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 ⊢ (𝐴 ⋉ 𝐶) = ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐶)) | |
4 | df-xrn 36646 | . 2 ⊢ (𝐵 ⋉ 𝐶) = ((◡(1st ↾ (V × V)) ∘ 𝐵) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐶)) | |
5 | 2, 3, 4 | 3eqtr4g 2801 | 1 ⊢ (𝐴 = 𝐵 → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶)) |
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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