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Mirrors > Home > MPE Home > Th. List > xpsfval | Structured version Visualization version GIF version |
Description: The value of the function appearing in xpsval 16629. (Contributed by Mario Carneiro, 15-Aug-2015.) |
Ref | Expression |
---|---|
xpsff1o.f | ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ◡({𝑥} +𝑐 {𝑦})) |
Ref | Expression |
---|---|
xpsfval | ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐹𝑌) = ◡({𝑋} +𝑐 {𝑌})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 4408 | . . . 4 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋}) | |
2 | sneq 4408 | . . . 4 ⊢ (𝑦 = 𝑌 → {𝑦} = {𝑌}) | |
3 | 1, 2 | oveqan12d 6943 | . . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ({𝑥} +𝑐 {𝑦}) = ({𝑋} +𝑐 {𝑌})) |
4 | 3 | cnveqd 5545 | . 2 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ◡({𝑥} +𝑐 {𝑦}) = ◡({𝑋} +𝑐 {𝑌})) |
5 | xpsff1o.f | . 2 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ◡({𝑥} +𝑐 {𝑦})) | |
6 | ovex 6956 | . . 3 ⊢ ({𝑋} +𝑐 {𝑌}) ∈ V | |
7 | 6 | cnvex 7394 | . 2 ⊢ ◡({𝑋} +𝑐 {𝑌}) ∈ V |
8 | 4, 5, 7 | ovmpt2a 7070 | 1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐹𝑌) = ◡({𝑋} +𝑐 {𝑌})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 {csn 4398 ◡ccnv 5356 (class class class)co 6924 ↦ cmpt2 6926 +𝑐 ccda 9326 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-iota 6101 df-fun 6139 df-fv 6145 df-ov 6927 df-oprab 6928 df-mpt2 6929 |
This theorem is referenced by: xpsff1o 16625 xpsaddlem 16632 xpsvsca 16636 xpsle 16638 xpsdsval 22605 |
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