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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrneq12 | Structured version Visualization version GIF version |
Description: Equality theorem for the range Cartesian product. (Contributed by Peter Mazsa, 16-Dec-2020.) |
Ref | Expression |
---|---|
xrneq12 | ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrneq1 35509 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶)) | |
2 | xrneq2 35512 | . 2 ⊢ (𝐶 = 𝐷 → (𝐵 ⋉ 𝐶) = (𝐵 ⋉ 𝐷)) | |
3 | 1, 2 | sylan9eq 2873 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ⋉ cxrn 35333 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-in 3940 df-ss 3949 df-br 5058 df-opab 5120 df-co 5557 df-xrn 35503 |
This theorem is referenced by: xrneq12i 35516 xrneq12d 35517 |
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