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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrneq12i | Structured version Visualization version GIF version |
Description: Equality theorem for the range Cartesian product, inference form. (Contributed by Peter Mazsa, 16-Dec-2020.) |
Ref | Expression |
---|---|
xrneq12i.1 | ⊢ 𝐴 = 𝐵 |
xrneq12i.2 | ⊢ 𝐶 = 𝐷 |
Ref | Expression |
---|---|
xrneq12i | ⊢ (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrneq12i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | xrneq12i.2 | . 2 ⊢ 𝐶 = 𝐷 | |
3 | xrneq12 37557 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐷)) | |
4 | 1, 2, 3 | mp2an 689 | 1 ⊢ (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 ⋉ cxrn 37346 |
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 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1543 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-rab 3432 df-v 3475 df-in 3955 df-ss 3965 df-br 5149 df-opab 5211 df-co 5685 df-xrn 37545 |
This theorem is referenced by: xrnres4 37579 xrnresex 37580 |
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