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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 5563 | . 2 ⊢ (𝐴 × 𝐵) = {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} | |
2 | nfxp.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | nfcri 2973 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 |
4 | nfxp.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
5 | 4 | nfcri 2973 | . . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐵 |
6 | 3, 5 | nfan 1900 | . . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) |
7 | 6 | nfopab 5136 | . 2 ⊢ Ⅎ𝑥{〈𝑦, 𝑧〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} |
8 | 1, 7 | nfcxfr 2977 | 1 ⊢ Ⅎ𝑥(𝐴 × 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 ∈ wcel 2114 Ⅎwnfc 2963 {copab 5130 × cxp 5555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-opab 5131 df-xp 5563 |
This theorem is referenced by: opeliunxp 5621 nfres 5857 mpomptsx 7764 dmmpossx 7766 fmpox 7767 ovmptss 7790 nfdju 9338 axcc2 9861 fsum2dlem 15127 fsumcom2 15131 fprod2dlem 15336 fprodcom2 15340 gsumcom2 19097 prdsdsf 22979 prdsxmet 22981 aciunf1lem 30409 esum2dlem 31353 poimirlem16 34910 poimirlem19 34913 dvnprodlem1 42238 stoweidlem21 42313 stoweidlem47 42339 opeliun2xp 44388 dmmpossx2 44392 |
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