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Mirrors > Home > ILE Home > Th. List > ovresd | GIF version |
Description: Lemma for converting metric theorems to metric space theorems. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
ovresd.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑋) |
ovresd.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑋) |
Ref | Expression |
---|---|
ovresd | ⊢ (𝜑 → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovresd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
2 | ovresd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑋) | |
3 | ovres 5972 | . 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵)) | |
4 | 1, 2, 3 | syl2anc 409 | 1 ⊢ (𝜑 → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ∈ wcel 2135 × cxp 4596 ↾ cres 4600 (class class class)co 5836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-xp 4604 df-res 4610 df-iota 5147 df-fv 5190 df-ov 5839 |
This theorem is referenced by: psmetres2 12880 xmetres2 12926 xmssym 13016 xmstri2 13017 mstri2 13018 xmstri 13019 mstri 13020 xmstri3 13021 mstri3 13022 msrtri 13023 limcimolemlt 13180 cnplimcim 13183 cnplimclemr 13185 limccnpcntop 13191 limccnp2lem 13192 |
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