Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ffoss | Structured version Visualization version 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 6437 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | |
2 | dffn4 6694 | . . . . 5 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴–onto→ran 𝐹) | |
3 | 2 | anbi1i 624 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) ↔ (𝐹:𝐴–onto→ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵)) |
4 | 1, 3 | bitri 274 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹:𝐴–onto→ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵)) |
5 | f11o.1 | . . . . 5 ⊢ 𝐹 ∈ V | |
6 | 5 | rnex 7759 | . . . 4 ⊢ ran 𝐹 ∈ V |
7 | foeq3 6686 | . . . . 5 ⊢ (𝑥 = ran 𝐹 → (𝐹:𝐴–onto→𝑥 ↔ 𝐹:𝐴–onto→ran 𝐹)) | |
8 | sseq1 3946 | . . . . 5 ⊢ (𝑥 = ran 𝐹 → (𝑥 ⊆ 𝐵 ↔ ran 𝐹 ⊆ 𝐵)) | |
9 | 7, 8 | anbi12d 631 | . . . 4 ⊢ (𝑥 = ran 𝐹 → ((𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ (𝐹:𝐴–onto→ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵))) |
10 | 6, 9 | spcev 3545 | . . 3 ⊢ ((𝐹:𝐴–onto→ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → ∃𝑥(𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵)) |
11 | 4, 10 | sylbi 216 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → ∃𝑥(𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵)) |
12 | fof 6688 | . . . 4 ⊢ (𝐹:𝐴–onto→𝑥 → 𝐹:𝐴⟶𝑥) | |
13 | fss 6617 | . . . 4 ⊢ ((𝐹:𝐴⟶𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝐹:𝐴⟶𝐵) | |
14 | 12, 13 | sylan 580 | . . 3 ⊢ ((𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝐹:𝐴⟶𝐵) |
15 | 14 | exlimiv 1933 | . 2 ⊢ (∃𝑥(𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝐹:𝐴⟶𝐵) |
16 | 11, 15 | impbii 208 | 1 ⊢ (𝐹:𝐴⟶𝐵 ↔ ∃𝑥(𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1539 ∃wex 1782 ∈ wcel 2106 Vcvv 3432 ⊆ wss 3887 ran crn 5590 Fn wfn 6428 ⟶wf 6429 –onto→wfo 6431 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-cnv 5597 df-dm 5599 df-rn 5600 df-f 6437 df-fo 6439 |
This theorem is referenced by: f11o 7789 |
Copyright terms: Public domain | W3C validator |