Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > dff2 | GIF version |
Description: Alternate definition of a mapping. (Contributed by NM, 14-Nov-2007.) |
Ref | Expression |
---|---|
dff2 | ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ 𝐹 ⊆ (𝐴 × 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffn 5267 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
2 | fssxp 5285 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵)) | |
3 | 1, 2 | jca 304 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 Fn 𝐴 ∧ 𝐹 ⊆ (𝐴 × 𝐵))) |
4 | rnss 4764 | . . . . 5 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹 ⊆ ran (𝐴 × 𝐵)) | |
5 | rnxpss 4965 | . . . . 5 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵 | |
6 | 4, 5 | sstrdi 3104 | . . . 4 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹 ⊆ 𝐵) |
7 | 6 | anim2i 339 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) → (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) |
8 | df-f 5122 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | |
9 | 7, 8 | sylibr 133 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) → 𝐹:𝐴⟶𝐵) |
10 | 3, 9 | impbii 125 | 1 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ 𝐹 ⊆ (𝐴 × 𝐵))) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ⊆ wss 3066 × cxp 4532 ran crn 4535 Fn wfn 5113 ⟶wf 5114 |
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-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-xp 4540 df-rel 4541 df-cnv 4542 df-dm 4544 df-rn 4545 df-fun 5120 df-fn 5121 df-f 5122 |
This theorem is referenced by: mapval2 6565 frecuzrdgtclt 10187 |
Copyright terms: Public domain | W3C validator |