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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 16835. (Contributed by Mario Carneiro, 15-Aug-2015.) |
Ref | Expression |
---|---|
xpsff1o.f | ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) |
Ref | Expression |
---|---|
xpsfval | ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐹𝑌) = {〈∅, 𝑋〉, 〈1o, 𝑌〉}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 486 | . . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋) | |
2 | 1 | opeq2d 4772 | . . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 〈∅, 𝑥〉 = 〈∅, 𝑋〉) |
3 | simpr 488 | . . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌) | |
4 | 3 | opeq2d 4772 | . . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 〈1o, 𝑦〉 = 〈1o, 𝑌〉) |
5 | 2, 4 | preq12d 4637 | . 2 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → {〈∅, 𝑥〉, 〈1o, 𝑦〉} = {〈∅, 𝑋〉, 〈1o, 𝑌〉}) |
6 | xpsff1o.f | . 2 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) | |
7 | prex 5298 | . 2 ⊢ {〈∅, 𝑋〉, 〈1o, 𝑌〉} ∈ V | |
8 | 5, 6, 7 | ovmpoa 7284 | 1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐹𝑌) = {〈∅, 𝑋〉, 〈1o, 𝑌〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∅c0 4243 {cpr 4527 〈cop 4531 (class class class)co 7135 ∈ cmpo 7137 1oc1o 8078 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 |
This theorem is referenced by: xpsff1o 16832 xpsaddlem 16838 xpsvsca 16842 xpsle 16844 xpsdsval 22988 |
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