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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 5873 | . 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {〈〈𝑥, 𝑦〉, 𝑤〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)} | |
2 | nfmpo.1 | . . . . . 6 ⊢ Ⅎ𝑧𝐴 | |
3 | 2 | nfcri 2313 | . . . . 5 ⊢ Ⅎ𝑧 𝑥 ∈ 𝐴 |
4 | nfmpo.2 | . . . . . 6 ⊢ Ⅎ𝑧𝐵 | |
5 | 4 | nfcri 2313 | . . . . 5 ⊢ Ⅎ𝑧 𝑦 ∈ 𝐵 |
6 | 3, 5 | nfan 1565 | . . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) |
7 | nfmpo.3 | . . . . 5 ⊢ Ⅎ𝑧𝐶 | |
8 | 7 | nfeq2 2331 | . . . 4 ⊢ Ⅎ𝑧 𝑤 = 𝐶 |
9 | 6, 8 | nfan 1565 | . . 3 ⊢ Ⅎ𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶) |
10 | 9 | nfoprab 5920 | . 2 ⊢ Ⅎ𝑧{〈〈𝑥, 𝑦〉, 𝑤〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)} |
11 | 1, 10 | nfcxfr 2316 | 1 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 = wceq 1353 ∈ wcel 2148 Ⅎwnfc 2306 {coprab 5869 ∈ cmpo 5870 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-oprab 5872 df-mpo 5873 |
This theorem is referenced by: nfof 6081 nfseq 10428 |
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