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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 38358 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶)) | |
2 | xrneq2 38361 | . 2 ⊢ (𝐶 = 𝐷 → (𝐵 ⋉ 𝐶) = (𝐵 ⋉ 𝐷)) | |
3 | 1, 2 | sylan9eq 2794 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ⋉ cxrn 38160 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3433 df-in 3969 df-ss 3979 df-br 5148 df-opab 5210 df-co 5697 df-xrn 38352 |
This theorem is referenced by: xrneq12i 38365 xrneq12d 38366 |
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