![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > dff3im | GIF version |
Description: Property of a mapping. (Contributed by Jim Kingdon, 4-Jan-2019.) |
Ref | Expression |
---|---|
dff3im | ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fssxp 5226 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵)) | |
2 | ffun 5211 | . . . . . . . 8 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹) | |
3 | 2 | adantr 272 | . . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → Fun 𝐹) |
4 | fdm 5214 | . . . . . . . . 9 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | |
5 | 4 | eleq2d 2169 | . . . . . . . 8 ⊢ (𝐹:𝐴⟶𝐵 → (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ 𝐴)) |
6 | 5 | biimpar 293 | . . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝐹) |
7 | funfvop 5464 | . . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → 〈𝑥, (𝐹‘𝑥)〉 ∈ 𝐹) | |
8 | 3, 6, 7 | syl2anc 406 | . . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 〈𝑥, (𝐹‘𝑥)〉 ∈ 𝐹) |
9 | df-br 3876 | . . . . . 6 ⊢ (𝑥𝐹(𝐹‘𝑥) ↔ 〈𝑥, (𝐹‘𝑥)〉 ∈ 𝐹) | |
10 | 8, 9 | sylibr 133 | . . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥𝐹(𝐹‘𝑥)) |
11 | funfvex 5370 | . . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ V) | |
12 | breq2 3879 | . . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑥𝐹𝑦 ↔ 𝑥𝐹(𝐹‘𝑥))) | |
13 | 12 | spcegv 2729 | . . . . . . 7 ⊢ ((𝐹‘𝑥) ∈ V → (𝑥𝐹(𝐹‘𝑥) → ∃𝑦 𝑥𝐹𝑦)) |
14 | 11, 13 | syl 14 | . . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝑥𝐹(𝐹‘𝑥) → ∃𝑦 𝑥𝐹𝑦)) |
15 | 3, 6, 14 | syl2anc 406 | . . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑥𝐹(𝐹‘𝑥) → ∃𝑦 𝑥𝐹𝑦)) |
16 | 10, 15 | mpd 13 | . . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 𝑥𝐹𝑦) |
17 | funmo 5074 | . . . . . 6 ⊢ (Fun 𝐹 → ∃*𝑦 𝑥𝐹𝑦) | |
18 | 2, 17 | syl 14 | . . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → ∃*𝑦 𝑥𝐹𝑦) |
19 | 18 | adantr 272 | . . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ∃*𝑦 𝑥𝐹𝑦) |
20 | eu5 2007 | . . . 4 ⊢ (∃!𝑦 𝑥𝐹𝑦 ↔ (∃𝑦 𝑥𝐹𝑦 ∧ ∃*𝑦 𝑥𝐹𝑦)) | |
21 | 16, 19, 20 | sylanbrc 411 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ∃!𝑦 𝑥𝐹𝑦) |
22 | 21 | ralrimiva 2464 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦) |
23 | 1, 22 | jca 302 | 1 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∃wex 1436 ∈ wcel 1448 ∃!weu 1960 ∃*wmo 1961 ∀wral 2375 Vcvv 2641 ⊆ wss 3021 〈cop 3477 class class class wbr 3875 × cxp 4475 dom cdm 4477 Fun wfun 5053 ⟶wf 5055 ‘cfv 5059 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-v 2643 df-sbc 2863 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-br 3876 df-opab 3930 df-id 4153 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-fv 5067 |
This theorem is referenced by: dff4im 5498 |
Copyright terms: Public domain | W3C validator |