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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 38378 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶)) | |
| 2 | xrneq2 38381 | . 2 ⊢ (𝐶 = 𝐷 → (𝐵 ⋉ 𝐶) = (𝐵 ⋉ 𝐷)) | |
| 3 | 1, 2 | sylan9eq 2797 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐷)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ⋉ 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-tru 1543 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: xrneq12i 38385 xrneq12d 38386 |
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