Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > ffoss | GIF version |
Description: Relationship between a mapping and an onto mapping. Figure 38 of [Enderton] p. 145. (Contributed by NM, 10-May-1998.) |
Ref | Expression |
---|---|
f11o.1 | ⊢ 𝐹 ∈ V |
Ref | Expression |
---|---|
ffoss | ⊢ (𝐹:𝐴⟶𝐵 ↔ ∃𝑥(𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-f 5187 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | |
2 | dffn4 5411 | . . . . 5 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴–onto→ran 𝐹) | |
3 | 2 | anbi1i 454 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) ↔ (𝐹:𝐴–onto→ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵)) |
4 | 1, 3 | bitri 183 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹:𝐴–onto→ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵)) |
5 | f11o.1 | . . . . 5 ⊢ 𝐹 ∈ V | |
6 | 5 | rnex 4866 | . . . 4 ⊢ ran 𝐹 ∈ V |
7 | foeq3 5403 | . . . . 5 ⊢ (𝑥 = ran 𝐹 → (𝐹:𝐴–onto→𝑥 ↔ 𝐹:𝐴–onto→ran 𝐹)) | |
8 | sseq1 3161 | . . . . 5 ⊢ (𝑥 = ran 𝐹 → (𝑥 ⊆ 𝐵 ↔ ran 𝐹 ⊆ 𝐵)) | |
9 | 7, 8 | anbi12d 465 | . . . 4 ⊢ (𝑥 = ran 𝐹 → ((𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ (𝐹:𝐴–onto→ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵))) |
10 | 6, 9 | spcev 2817 | . . 3 ⊢ ((𝐹:𝐴–onto→ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → ∃𝑥(𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵)) |
11 | 4, 10 | sylbi 120 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → ∃𝑥(𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵)) |
12 | fof 5405 | . . . 4 ⊢ (𝐹:𝐴–onto→𝑥 → 𝐹:𝐴⟶𝑥) | |
13 | fss 5344 | . . . 4 ⊢ ((𝐹:𝐴⟶𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝐹:𝐴⟶𝐵) | |
14 | 12, 13 | sylan 281 | . . 3 ⊢ ((𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝐹:𝐴⟶𝐵) |
15 | 14 | exlimiv 1585 | . 2 ⊢ (∃𝑥(𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝐹:𝐴⟶𝐵) |
16 | 11, 15 | impbii 125 | 1 ⊢ (𝐹:𝐴⟶𝐵 ↔ ∃𝑥(𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1342 ∃wex 1479 ∈ wcel 2135 Vcvv 2722 ⊆ wss 3112 ran crn 4600 Fn wfn 5178 ⟶wf 5179 –onto→wfo 5181 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4095 ax-pow 4148 ax-pr 4182 ax-un 4406 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-rex 2448 df-v 2724 df-un 3116 df-in 3118 df-ss 3125 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-br 3978 df-opab 4039 df-cnv 4607 df-dm 4609 df-rn 4610 df-f 5187 df-fo 5189 |
This theorem is referenced by: f11o 5460 |
Copyright terms: Public domain | W3C validator |