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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 5683 | . 2 ⊢ (𝐴 × 𝐵) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} | |
2 | nfxp.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | nfcri 2891 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 |
4 | nfxp.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
5 | 4 | nfcri 2891 | . . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐵 |
6 | 3, 5 | nfan 1903 | . . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) |
7 | 6 | nfopab 5218 | . 2 ⊢ Ⅎ𝑥{⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} |
8 | 1, 7 | nfcxfr 2902 | 1 ⊢ Ⅎ𝑥(𝐴 × 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 ∈ wcel 2107 Ⅎwnfc 2884 {copab 5211 × cxp 5675 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-opab 5212 df-xp 5683 |
This theorem is referenced by: opeliunxp 5744 nfres 5984 mpomptsx 8050 dmmpossx 8052 fmpox 8053 ovmptss 8079 nfdju 9902 axcc2 10432 fsum2dlem 15716 fsumcom2 15720 fprod2dlem 15924 fprodcom2 15928 gsumcom2 19843 prdsdsf 23873 prdsxmet 23875 djussxp2 31873 aciunf1lem 31887 gsumpart 32207 esum2dlem 33090 poimirlem16 36504 poimirlem19 36507 dvnprodlem1 44662 stoweidlem21 44737 stoweidlem47 44763 opeliun2xp 47008 dmmpossx2 47012 |
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