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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 35650 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐷)) | |
4 | 1, 2, 3 | mp2an 690 | 1 ⊢ (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ⋉ cxrn 35467 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-in 3943 df-ss 3952 df-br 5067 df-opab 5129 df-co 5564 df-xrn 35638 |
This theorem is referenced by: xrnres4 35668 xrnresex 35669 |
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