![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > xpsfval | Structured version Visualization version GIF version |
Description: The value of the function appearing in xpsval 17580. (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 481 | . . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋) | |
2 | 1 | opeq2d 4878 | . . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 〈∅, 𝑥〉 = 〈∅, 𝑋〉) |
3 | simpr 483 | . . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌) | |
4 | 3 | opeq2d 4878 | . . 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 5430 | . 2 ⊢ {〈∅, 𝑋〉, 〈1o, 𝑌〉} ∈ V | |
8 | 5, 6, 7 | ovmpoa 7573 | 1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐹𝑌) = {〈∅, 𝑋〉, 〈1o, 𝑌〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∅c0 4322 {cpr 4625 〈cop 4629 (class class class)co 7416 ∈ cmpo 7418 1oc1o 8481 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pr 5425 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3776 df-dif 3949 df-un 3951 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-iota 6498 df-fun 6548 df-fv 6554 df-ov 7419 df-oprab 7420 df-mpo 7421 |
This theorem is referenced by: xpsff1o 17577 xpsaddlem 17583 xpsvsca 17587 xpsle 17589 xpsdsval 24375 |
Copyright terms: Public domain | W3C validator |