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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 16837. (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 485 | . . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋) | |
2 | 1 | opeq2d 4803 | . . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 〈∅, 𝑥〉 = 〈∅, 𝑋〉) |
3 | simpr 487 | . . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌) | |
4 | 3 | opeq2d 4803 | . . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 〈1o, 𝑦〉 = 〈1o, 𝑌〉) |
5 | 2, 4 | preq12d 4670 | . 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 5324 | . 2 ⊢ {〈∅, 𝑋〉, 〈1o, 𝑌〉} ∈ V | |
8 | 5, 6, 7 | ovmpoa 7299 | 1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐹𝑌) = {〈∅, 𝑋〉, 〈1o, 𝑌〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∅c0 4290 {cpr 4562 〈cop 4566 (class class class)co 7150 ∈ cmpo 7152 1oc1o 8089 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-iota 6308 df-fun 6351 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 |
This theorem is referenced by: xpsff1o 16834 xpsaddlem 16840 xpsvsca 16844 xpsle 16846 xpsdsval 22985 |
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