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Mirrors > Home > ILE Home > Th. List > nfmpt2 | GIF version |
Description: Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.) |
Ref | Expression |
---|---|
nfmpt2.1 | ⊢ Ⅎ𝑧𝐴 |
nfmpt2.2 | ⊢ Ⅎ𝑧𝐵 |
nfmpt2.3 | ⊢ Ⅎ𝑧𝐶 |
Ref | Expression |
---|---|
nfmpt2 | ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mpt2 5517 | . 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {〈〈𝑥, 𝑦〉, 𝑤〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)} | |
2 | nfmpt2.1 | . . . . . 6 ⊢ Ⅎ𝑧𝐴 | |
3 | 2 | nfcri 2172 | . . . . 5 ⊢ Ⅎ𝑧 𝑥 ∈ 𝐴 |
4 | nfmpt2.2 | . . . . . 6 ⊢ Ⅎ𝑧𝐵 | |
5 | 4 | nfcri 2172 | . . . . 5 ⊢ Ⅎ𝑧 𝑦 ∈ 𝐵 |
6 | 3, 5 | nfan 1457 | . . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) |
7 | nfmpt2.3 | . . . . 5 ⊢ Ⅎ𝑧𝐶 | |
8 | 7 | nfeq2 2189 | . . . 4 ⊢ Ⅎ𝑧 𝑤 = 𝐶 |
9 | 6, 8 | nfan 1457 | . . 3 ⊢ Ⅎ𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶) |
10 | 9 | nfoprab 5557 | . 2 ⊢ Ⅎ𝑧{〈〈𝑥, 𝑦〉, 𝑤〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)} |
11 | 1, 10 | nfcxfr 2175 | 1 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 = wceq 1243 ∈ wcel 1393 Ⅎwnfc 2165 {coprab 5513 ↦ cmpt2 5514 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-oprab 5516 df-mpt2 5517 |
This theorem is referenced by: nfiseq 9218 |
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