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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 16846. (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 4813 | . . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 〈∅, 𝑥〉 = 〈∅, 𝑋〉) |
3 | simpr 487 | . . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌) | |
4 | 3 | opeq2d 4813 | . . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 〈1o, 𝑦〉 = 〈1o, 𝑌〉) |
5 | 2, 4 | preq12d 4680 | . 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 5336 | . 2 ⊢ {〈∅, 𝑋〉, 〈1o, 𝑌〉} ∈ V | |
8 | 5, 6, 7 | ovmpoa 7308 | 1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐹𝑌) = {〈∅, 𝑋〉, 〈1o, 𝑌〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∅c0 4294 {cpr 4572 〈cop 4576 (class class class)co 7159 ∈ cmpo 7161 1oc1o 8098 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-iota 6317 df-fun 6360 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 |
This theorem is referenced by: xpsff1o 16843 xpsaddlem 16849 xpsvsca 16853 xpsle 16855 xpsdsval 22994 |
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