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| Description: Alternate definition of a mapping. (Contributed by NM, 14-Nov-2007.) | 
| Ref | Expression | 
|---|---|
| dff2 | ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ 𝐹 ⊆ (𝐴 × 𝐵))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ffn 6735 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
| 2 | fssxp 6762 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵)) | |
| 3 | 1, 2 | jca 511 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 Fn 𝐴 ∧ 𝐹 ⊆ (𝐴 × 𝐵))) | 
| 4 | rnss 5949 | . . . . 5 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹 ⊆ ran (𝐴 × 𝐵)) | |
| 5 | rnxpss 6191 | . . . . 5 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵 | |
| 6 | 4, 5 | sstrdi 3995 | . . . 4 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹 ⊆ 𝐵) | 
| 7 | 6 | anim2i 617 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) → (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | 
| 8 | df-f 6564 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | |
| 9 | 7, 8 | sylibr 234 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) → 𝐹:𝐴⟶𝐵) | 
| 10 | 3, 9 | impbii 209 | 1 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ 𝐹 ⊆ (𝐴 × 𝐵))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 ⊆ wss 3950 × cxp 5682 ran crn 5685 Fn wfn 6555 ⟶wf 6556 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 df-rel 5691 df-cnv 5692 df-dm 5694 df-rn 5695 df-fun 6562 df-fn 6563 df-f 6564 | 
| This theorem is referenced by: fpr2g 7232 mapval2 8913 cardf2 9984 mpoaddf 11250 mpomulf 11251 imasaddflem 17576 imasvscaf 17585 gsumpart 33061 | 
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