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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 6388 | . . . . . 6 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹) | |
2 | 1 | biimpi 218 | . . . . 5 ⊢ (Fun 𝐹 → 𝐹 Fn dom 𝐹) |
3 | rnss 5812 | . . . . . 6 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹 ⊆ ran (𝐴 × 𝐵)) | |
4 | rnxpss 6032 | . . . . . 6 ⊢ ran (𝐴 × 𝐵) ⊆ 𝐵 | |
5 | 3, 4 | sstrdi 3982 | . . . . 5 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹 ⊆ 𝐵) |
6 | 2, 5 | anim12i 614 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) → (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ 𝐵)) |
7 | df-f 6362 | . . . 4 ⊢ (𝐹:dom 𝐹⟶𝐵 ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ 𝐵)) | |
8 | 6, 7 | sylibr 236 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) → 𝐹:dom 𝐹⟶𝐵) |
9 | dmss 5774 | . . . . 5 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → dom 𝐹 ⊆ dom (𝐴 × 𝐵)) | |
10 | dmxpss 6031 | . . . . 5 ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴 | |
11 | 9, 10 | sstrdi 3982 | . . . 4 ⊢ (𝐹 ⊆ (𝐴 × 𝐵) → dom 𝐹 ⊆ 𝐴) |
12 | 11 | adantl 484 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) → dom 𝐹 ⊆ 𝐴) |
13 | 8, 12 | jca 514 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) → (𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴)) |
14 | ffun 6520 | . . . 4 ⊢ (𝐹:dom 𝐹⟶𝐵 → Fun 𝐹) | |
15 | 14 | adantr 483 | . . 3 ⊢ ((𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴) → Fun 𝐹) |
16 | fssxp 6537 | . . . 4 ⊢ (𝐹:dom 𝐹⟶𝐵 → 𝐹 ⊆ (dom 𝐹 × 𝐵)) | |
17 | xpss1 5577 | . . . 4 ⊢ (dom 𝐹 ⊆ 𝐴 → (dom 𝐹 × 𝐵) ⊆ (𝐴 × 𝐵)) | |
18 | 16, 17 | sylan9ss 3983 | . . 3 ⊢ ((𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴) → 𝐹 ⊆ (𝐴 × 𝐵)) |
19 | 15, 18 | jca 514 | . 2 ⊢ ((𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴) → (Fun 𝐹 ∧ 𝐹 ⊆ (𝐴 × 𝐵))) |
20 | 13, 19 | impbii 211 | 1 ⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) ↔ (𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ⊆ wss 3939 × cxp 5556 dom cdm 5558 ran crn 5559 Fun wfun 6352 Fn wfn 6353 ⟶wf 6354 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-br 5070 df-opab 5132 df-xp 5564 df-rel 5565 df-cnv 5566 df-dm 5568 df-rn 5569 df-fun 6360 df-fn 6361 df-f 6362 |
This theorem is referenced by: elpm2g 8426 volf 24133 |
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