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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 6598 | . . . . . 6 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹) | |
2 | 1 | biimpi 216 | . . . . 5 ⊢ (Fun 𝐹 → 𝐹 Fn dom 𝐹) |
3 | rnss 5953 | . . . . . 6 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹 ⊆ ran (𝐴 × 𝐵)) | |
4 | rnxpss 6194 | . . . . . 6 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵 | |
5 | 3, 4 | sstrdi 4008 | . . . . 5 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹 ⊆ 𝐵) |
6 | 2, 5 | anim12i 613 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) → (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ 𝐵)) |
7 | df-f 6567 | . . . 4 ⊢ (𝐹:dom 𝐹⟶𝐵 ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ 𝐵)) | |
8 | 6, 7 | sylibr 234 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) → 𝐹:dom 𝐹⟶𝐵) |
9 | dmss 5916 | . . . . 5 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → dom 𝐹 ⊆ dom (𝐴 × 𝐵)) | |
10 | dmxpss 6193 | . . . . 5 ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴 | |
11 | 9, 10 | sstrdi 4008 | . . . 4 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → dom 𝐹 ⊆ 𝐴) |
12 | 11 | adantl 481 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) → dom 𝐹 ⊆ 𝐴) |
13 | 8, 12 | jca 511 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) → (𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴)) |
14 | ffun 6740 | . . . 4 ⊢ (𝐹:dom 𝐹⟶𝐵 → Fun 𝐹) | |
15 | 14 | adantr 480 | . . 3 ⊢ ((𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴) → Fun 𝐹) |
16 | fssxp 6764 | . . . 4 ⊢ (𝐹:dom 𝐹⟶𝐵 → 𝐹 ⊆ (dom 𝐹 × 𝐵)) | |
17 | xpss1 5708 | . . . 4 ⊢ (dom 𝐹 ⊆ 𝐴 → (dom 𝐹 × 𝐵) ⊆ (𝐴 × 𝐵)) | |
18 | 16, 17 | sylan9ss 4009 | . . 3 ⊢ ((𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴) → 𝐹 ⊆ (𝐴 × 𝐵)) |
19 | 15, 18 | jca 511 | . 2 ⊢ ((𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴) → (Fun 𝐹 ∧ 𝐹 ⊆ (𝐴 × 𝐵))) |
20 | 13, 19 | impbii 209 | 1 ⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) ↔ (𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 ⊆ wss 3963 × cxp 5687 dom cdm 5689 ran crn 5690 Fun wfun 6557 Fn wfn 6558 ⟶wf 6559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-dm 5699 df-rn 5700 df-fun 6565 df-fn 6566 df-f 6567 |
This theorem is referenced by: elpm2g 8883 volf 25578 dfno2 43418 |
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