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Mirrors > Home > ILE Home > Th. List > nfmpo | GIF version |
Description: Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.) |
Ref | Expression |
---|---|
nfmpo.1 | ⊢ Ⅎ𝑧𝐴 |
nfmpo.2 | ⊢ Ⅎ𝑧𝐵 |
nfmpo.3 | ⊢ Ⅎ𝑧𝐶 |
Ref | Expression |
---|---|
nfmpo | ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mpo 5841 | . 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {〈〈𝑥, 𝑦〉, 𝑤〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)} | |
2 | nfmpo.1 | . . . . . 6 ⊢ Ⅎ𝑧𝐴 | |
3 | 2 | nfcri 2300 | . . . . 5 ⊢ Ⅎ𝑧 𝑥 ∈ 𝐴 |
4 | nfmpo.2 | . . . . . 6 ⊢ Ⅎ𝑧𝐵 | |
5 | 4 | nfcri 2300 | . . . . 5 ⊢ Ⅎ𝑧 𝑦 ∈ 𝐵 |
6 | 3, 5 | nfan 1552 | . . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) |
7 | nfmpo.3 | . . . . 5 ⊢ Ⅎ𝑧𝐶 | |
8 | 7 | nfeq2 2318 | . . . 4 ⊢ Ⅎ𝑧 𝑤 = 𝐶 |
9 | 6, 8 | nfan 1552 | . . 3 ⊢ Ⅎ𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶) |
10 | 9 | nfoprab 5885 | . 2 ⊢ Ⅎ𝑧{〈〈𝑥, 𝑦〉, 𝑤〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)} |
11 | 1, 10 | nfcxfr 2303 | 1 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1342 ∈ wcel 2135 Ⅎwnfc 2293 {coprab 5837 ∈ cmpo 5838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-oprab 5840 df-mpo 5841 |
This theorem is referenced by: nfof 6049 nfseq 10380 |
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