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Mirrors > Home > MPE Home > Th. List > dff2 | Structured version Visualization version GIF version |
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 6338 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
2 | fssxp 6357 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵)) | |
3 | 1, 2 | jca 504 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 Fn 𝐴 ∧ 𝐹 ⊆ (𝐴 × 𝐵))) |
4 | rnss 5645 | . . . . 5 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹 ⊆ ran (𝐴 × 𝐵)) | |
5 | rnxpss 5863 | . . . . 5 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵 | |
6 | 4, 5 | syl6ss 3866 | . . . 4 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹 ⊆ 𝐵) |
7 | 6 | anim2i 607 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) → (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) |
8 | df-f 6186 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | |
9 | 7, 8 | sylibr 226 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) → 𝐹:𝐴⟶𝐵) |
10 | 3, 9 | impbii 201 | 1 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ 𝐹 ⊆ (𝐴 × 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 387 ⊆ wss 3825 × cxp 5398 ran crn 5401 Fn wfn 6177 ⟶wf 6178 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-sep 5054 ax-nul 5061 ax-pr 5180 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-br 4924 df-opab 4986 df-xp 5406 df-rel 5407 df-cnv 5408 df-dm 5410 df-rn 5411 df-fun 6184 df-fn 6185 df-f 6186 |
This theorem is referenced by: fpr2g 6794 mapval2 8228 cardf2 9158 imasaddflem 16649 imasvscaf 16658 |
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