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| 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 6565 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | |
| 2 | dffn4 6826 | . . . . 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 275 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹:𝐴–onto→ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵)) |
| 5 | f11o.1 | . . . . 5 ⊢ 𝐹 ∈ V | |
| 6 | 5 | rnex 7932 | . . . 4 ⊢ ran 𝐹 ∈ V |
| 7 | foeq3 6818 | . . . . 5 ⊢ (𝑥 = ran 𝐹 → (𝐹:𝐴–onto→𝑥 ↔ 𝐹:𝐴–onto→ran 𝐹)) | |
| 8 | sseq1 4009 | . . . . 5 ⊢ (𝑥 = ran 𝐹 → (𝑥 ⊆ 𝐵 ↔ ran 𝐹 ⊆ 𝐵)) | |
| 9 | 7, 8 | anbi12d 632 | . . . 4 ⊢ (𝑥 = ran 𝐹 → ((𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ (𝐹:𝐴–onto→ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵))) |
| 10 | 6, 9 | spcev 3606 | . . 3 ⊢ ((𝐹:𝐴–onto→ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → ∃𝑥(𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵)) |
| 11 | 4, 10 | sylbi 217 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → ∃𝑥(𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵)) |
| 12 | fof 6820 | . . . 4 ⊢ (𝐹:𝐴–onto→𝑥 → 𝐹:𝐴⟶𝑥) | |
| 13 | fss 6752 | . . . 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 1930 | . 2 ⊢ (∃𝑥(𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝐹:𝐴⟶𝐵) |
| 16 | 11, 15 | impbii 209 | 1 ⊢ (𝐹:𝐴⟶𝐵 ↔ ∃𝑥(𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 ran crn 5686 Fn wfn 6556 ⟶wf 6557 –onto→wfo 6559 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-cnv 5693 df-dm 5695 df-rn 5696 df-f 6565 df-fo 6567 |
| This theorem is referenced by: f11o 7971 |
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