![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > nfxp | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for Cartesian product. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Ref | Expression |
---|---|
nfxp.1 | ⊢ Ⅎ𝑥𝐴 |
nfxp.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
nfxp | ⊢ Ⅎ𝑥(𝐴 × 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-xp 5687 | . 2 ⊢ (𝐴 × 𝐵) = {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} | |
2 | nfxp.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | nfcri 2882 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 |
4 | nfxp.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
5 | 4 | nfcri 2882 | . . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐵 |
6 | 3, 5 | nfan 1894 | . . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) |
7 | 6 | nfopab 5221 | . 2 ⊢ Ⅎ𝑥{〈𝑦, 𝑧〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} |
8 | 1, 7 | nfcxfr 2889 | 1 ⊢ Ⅎ𝑥(𝐴 × 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 ∈ wcel 2098 Ⅎwnfc 2875 {copab 5214 × cxp 5679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-opab 5215 df-xp 5687 |
This theorem is referenced by: opeliunxp 5748 nfres 5990 mpomptsx 8077 dmmpossx 8079 fmpox 8080 ovmptss 8106 nfdju 9946 axcc2 10476 fsum2dlem 15769 fsumcom2 15773 fprod2dlem 15977 fprodcom2 15981 gsumcom2 19968 prdsdsf 24356 prdsxmet 24358 djussxp2 32556 aciunf1lem 32570 gsumpart 32901 esum2dlem 33881 poimirlem16 37285 poimirlem19 37288 dvnprodlem1 45504 stoweidlem21 45579 stoweidlem47 45605 opeliun2xp 47648 dmmpossx2 47652 |
Copyright terms: Public domain | W3C validator |