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 5044 | . 2 ⊢ (𝐴 × 𝐵) = {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} | |
2 | nfxp.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | nfcri 2745 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 |
4 | nfxp.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
5 | 4 | nfcri 2745 | . . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐵 |
6 | 3, 5 | nfan 1816 | . . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) |
7 | 6 | nfopab 4650 | . 2 ⊢ Ⅎ𝑥{〈𝑦, 𝑧〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} |
8 | 1, 7 | nfcxfr 2749 | 1 ⊢ Ⅎ𝑥(𝐴 × 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 ∈ wcel 1977 Ⅎwnfc 2738 {copab 4642 × cxp 5036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-opab 4644 df-xp 5044 |
This theorem is referenced by: opeliunxp 5093 nfres 5319 mpt2mptsx 7122 dmmpt2ssx 7124 fmpt2x 7125 ovmptss 7145 axcc2 9142 fsum2dlem 14343 fsumcom2 14347 fsumcom2OLD 14348 fprod2dlem 14549 fprodcom2 14553 fprodcom2OLD 14554 gsumcom2 18197 prdsdsf 21982 prdsxmet 21984 aciunf1lem 28844 esum2dlem 29481 poimirlem16 32595 poimirlem19 32598 dvnprodlem1 38836 stoweidlem21 38914 stoweidlem47 38940 opeliun2xp 41904 dmmpt2ssx2 41908 |
Copyright terms: Public domain | W3C validator |