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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 3986 | . 2 ⊢ (𝑦 ∈ 𝐴 ↦ 𝐵) = {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐵)} | |
2 | nfmpt.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | nfcri 2273 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 |
4 | nfmpt.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
5 | 4 | nfeq2 2291 | . . . 4 ⊢ Ⅎ𝑥 𝑧 = 𝐵 |
6 | 3, 5 | nfan 1544 | . . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐵) |
7 | 6 | nfopab 3991 | . 2 ⊢ Ⅎ𝑥{〈𝑦, 𝑧〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐵)} |
8 | 1, 7 | nfcxfr 2276 | 1 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ↦ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1331 ∈ wcel 1480 Ⅎwnfc 2266 {copab 3983 ↦ cmpt 3984 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-opab 3985 df-mpt 3986 |
This theorem is referenced by: nfof 5980 nffrec 6286 mapxpen 6735 nfsum1 11118 nfsum 11119 nfcprod1 11316 nfcprod 11317 ctiunct 11942 fsumcncntop 12714 limcmpted 12790 |
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