Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > nfmpt | GIF version |
Description: Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.) |
Ref | Expression |
---|---|
nfmpt.1 | ⊢ Ⅎ𝑥𝐴 |
nfmpt.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
nfmpt | ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ↦ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mpt 4039 | . 2 ⊢ (𝑦 ∈ 𝐴 ↦ 𝐵) = {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐵)} | |
2 | nfmpt.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | nfcri 2300 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 |
4 | nfmpt.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
5 | 4 | nfeq2 2318 | . . . 4 ⊢ Ⅎ𝑥 𝑧 = 𝐵 |
6 | 3, 5 | nfan 1552 | . . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐵) |
7 | 6 | nfopab 4044 | . 2 ⊢ Ⅎ𝑥{〈𝑦, 𝑧〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐵)} |
8 | 1, 7 | nfcxfr 2303 | 1 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ↦ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1342 ∈ wcel 2135 Ⅎwnfc 2293 {copab 4036 ↦ cmpt 4037 |
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-opab 4038 df-mpt 4039 |
This theorem is referenced by: nfof 6049 nffrec 6355 mapxpen 6805 nfsum1 11283 nfsum 11284 nfcprod1 11481 nfcprod 11482 ctiunct 12316 fsumcncntop 13103 limcmpted 13179 |
Copyright terms: Public domain | W3C validator |