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Mirrors > Home > ILE Home > Th. List > nfxp | GIF version |
Description: Bound-variable hypothesis builder for cross 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 4650 | . 2 ⊢ (𝐴 × 𝐵) = {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} | |
2 | nfxp.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | nfcri 2326 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 |
4 | nfxp.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
5 | 4 | nfcri 2326 | . . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐵 |
6 | 3, 5 | nfan 1576 | . . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) |
7 | 6 | nfopab 4086 | . 2 ⊢ Ⅎ𝑥{〈𝑦, 𝑧〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)} |
8 | 1, 7 | nfcxfr 2329 | 1 ⊢ Ⅎ𝑥(𝐴 × 𝐵) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ∈ wcel 2160 Ⅎwnfc 2319 {copab 4078 × cxp 4642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-opab 4080 df-xp 4650 |
This theorem is referenced by: opeliunxp 4699 nfres 4927 mpomptsx 6222 dmmpossx 6224 fmpox 6225 disjxp1 6261 nfdju 7071 fsum2dlemstep 11474 fisumcom2 11478 fprod2dlemstep 11662 fprodcom2fi 11666 |
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