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| Description: The value of the function appearing in xpsval 17616. (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 482 | . . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋) | |
| 2 | 1 | opeq2d 4879 | . . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 〈∅, 𝑥〉 = 〈∅, 𝑋〉) | 
| 3 | simpr 484 | . . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌) | |
| 4 | 3 | opeq2d 4879 | . . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 〈1o, 𝑦〉 = 〈1o, 𝑌〉) | 
| 5 | 2, 4 | preq12d 4740 | . 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 5436 | . 2 ⊢ {〈∅, 𝑋〉, 〈1o, 𝑌〉} ∈ V | |
| 8 | 5, 6, 7 | ovmpoa 7589 | 1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐹𝑌) = {〈∅, 𝑋〉, 〈1o, 𝑌〉}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∅c0 4332 {cpr 4627 〈cop 4631 (class class class)co 7432 ∈ cmpo 7434 1oc1o 8500 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-iota 6513 df-fun 6562 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 | 
| This theorem is referenced by: xpsff1o 17613 xpsaddlem 17619 xpsvsca 17623 xpsle 17625 xpsdsval 24392 | 
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