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Mirrors > Home > MPE Home > Th. List > funssxp | Structured version Visualization version GIF version |
Description: Two ways of specifying a partial function from 𝐴 to 𝐵. (Contributed by NM, 13-Nov-2007.) |
Ref | Expression |
---|---|
funssxp | ⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) ↔ (𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funfn 6131 | . . . . . 6 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹) | |
2 | 1 | biimpi 208 | . . . . 5 ⊢ (Fun 𝐹 → 𝐹 Fn dom 𝐹) |
3 | rnss 5557 | . . . . . 6 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹 ⊆ ran (𝐴 × 𝐵)) | |
4 | rnxpss 5783 | . . . . . 6 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵 | |
5 | 3, 4 | syl6ss 3810 | . . . . 5 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹 ⊆ 𝐵) |
6 | 2, 5 | anim12i 607 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) → (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ 𝐵)) |
7 | df-f 6105 | . . . 4 ⊢ (𝐹:dom 𝐹⟶𝐵 ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ 𝐵)) | |
8 | 6, 7 | sylibr 226 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) → 𝐹:dom 𝐹⟶𝐵) |
9 | dmss 5526 | . . . . 5 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → dom 𝐹 ⊆ dom (𝐴 × 𝐵)) | |
10 | dmxpss 5782 | . . . . 5 ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴 | |
11 | 9, 10 | syl6ss 3810 | . . . 4 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → dom 𝐹 ⊆ 𝐴) |
12 | 11 | adantl 474 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) → dom 𝐹 ⊆ 𝐴) |
13 | 8, 12 | jca 508 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) → (𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴)) |
14 | ffun 6259 | . . . 4 ⊢ (𝐹:dom 𝐹⟶𝐵 → Fun 𝐹) | |
15 | 14 | adantr 473 | . . 3 ⊢ ((𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴) → Fun 𝐹) |
16 | fssxp 6275 | . . . 4 ⊢ (𝐹:dom 𝐹⟶𝐵 → 𝐹 ⊆ (dom 𝐹 × 𝐵)) | |
17 | xpss1 5331 | . . . 4 ⊢ (dom 𝐹 ⊆ 𝐴 → (dom 𝐹 × 𝐵) ⊆ (𝐴 × 𝐵)) | |
18 | 16, 17 | sylan9ss 3811 | . . 3 ⊢ ((𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴) → 𝐹 ⊆ (𝐴 × 𝐵)) |
19 | 15, 18 | jca 508 | . 2 ⊢ ((𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴) → (Fun 𝐹 ∧ 𝐹 ⊆ (𝐴 × 𝐵))) |
20 | 13, 19 | impbii 201 | 1 ⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) ↔ (𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 385 ⊆ wss 3769 × cxp 5310 dom cdm 5312 ran crn 5313 Fun wfun 6095 Fn wfn 6096 ⟶wf 6097 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pr 5097 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-br 4844 df-opab 4906 df-xp 5318 df-rel 5319 df-cnv 5320 df-dm 5322 df-rn 5323 df-fun 6103 df-fn 6104 df-f 6105 |
This theorem is referenced by: elpm2g 8112 volf 23637 |
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