![]() |
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 5402 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵)) | |
2 | ffun 5387 | . . . . . . . 8 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹) | |
3 | 2 | adantr 276 | . . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → Fun 𝐹) |
4 | fdm 5390 | . . . . . . . . 9 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | |
5 | 4 | eleq2d 2259 | . . . . . . . 8 ⊢ (𝐹:𝐴⟶𝐵 → (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ 𝐴)) |
6 | 5 | biimpar 297 | . . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ dom 𝐹) |
7 | funfvop 5649 | . . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → 〈𝑥, (𝐹‘𝑥)〉 ∈ 𝐹) | |
8 | 3, 6, 7 | syl2anc 411 | . . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 〈𝑥, (𝐹‘𝑥)〉 ∈ 𝐹) |
9 | df-br 4019 | . . . . . 6 ⊢ (𝑥𝐹(𝐹‘𝑥) ↔ 〈𝑥, (𝐹‘𝑥)〉 ∈ 𝐹) | |
10 | 8, 9 | sylibr 134 | . . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥𝐹(𝐹‘𝑥)) |
11 | funfvex 5551 | . . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ V) | |
12 | breq2 4022 | . . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑥𝐹𝑦 ↔ 𝑥𝐹(𝐹‘𝑥))) | |
13 | 12 | spcegv 2840 | . . . . . . 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 411 | . . . . 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 5250 | . . . . . 6 ⊢ (Fun 𝐹 → ∃*𝑦 𝑥𝐹𝑦) | |
18 | 2, 17 | syl 14 | . . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → ∃*𝑦 𝑥𝐹𝑦) |
19 | 18 | adantr 276 | . . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ∃*𝑦 𝑥𝐹𝑦) |
20 | eu5 2085 | . . . 4 ⊢ (∃!𝑦 𝑥𝐹𝑦 ↔ (∃𝑦 𝑥𝐹𝑦 ∧ ∃*𝑦 𝑥𝐹𝑦)) | |
21 | 16, 19, 20 | sylanbrc 417 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ∃!𝑦 𝑥𝐹𝑦) |
22 | 21 | ralrimiva 2563 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦) |
23 | 1, 22 | jca 306 | 1 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∃wex 1503 ∃!weu 2038 ∃*wmo 2039 ∈ wcel 2160 ∀wral 2468 Vcvv 2752 ⊆ wss 3144 〈cop 3610 class class class wbr 4018 × cxp 4642 dom cdm 4644 Fun wfun 5229 ⟶wf 5231 ‘cfv 5235 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-sbc 2978 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-fv 5243 |
This theorem is referenced by: dff4im 5683 |
Copyright terms: Public domain | W3C validator |