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Mirrors > Home > ILE Home > Th. List > nfmpt1 | GIF version |
Description: Bound-variable hypothesis builder for the maps-to notation. (Contributed by FL, 17-Feb-2008.) |
Ref | Expression |
---|---|
nfmpt1 | ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mpt 3986 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑧〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)} | |
2 | nfopab1 3992 | . 2 ⊢ Ⅎ𝑥{〈𝑥, 𝑧〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)} | |
3 | 1, 2 | nfcxfr 2276 | 1 ⊢ Ⅎ𝑥(𝑥 ∈ 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-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: nffvmpt1 5425 fvmptss2 5489 fvmptssdm 5498 fvmptdf 5501 mpteqb 5504 fvmptf 5506 ralrnmpt 5555 rexrnmpt 5556 f1ompt 5564 f1mpt 5665 fliftfun 5690 dom2lem 6659 mapxpen 6735 mkvprop 7025 nfcprod1 11316 cnmpt11 12441 |
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