MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  funssxp Structured version   Visualization version   GIF version

Theorem funssxp 6764
Description: Two ways of specifying a partial function from 𝐴 to 𝐵. (Contributed by NM, 13-Nov-2007.)
Assertion
Ref Expression
funssxp ((Fun 𝐹𝐹 ⊆ (𝐴 × 𝐵)) ↔ (𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴))

Proof of Theorem funssxp
StepHypRef Expression
1 funfn 6596 . . . . . 6 (Fun 𝐹𝐹 Fn dom 𝐹)
21biimpi 216 . . . . 5 (Fun 𝐹𝐹 Fn dom 𝐹)
3 rnss 5950 . . . . . 6 (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹 ⊆ ran (𝐴 × 𝐵))
4 rnxpss 6192 . . . . . 6 ran (𝐴 × 𝐵) ⊆ 𝐵
53, 4sstrdi 3996 . . . . 5 (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹𝐵)
62, 5anim12i 613 . . . 4 ((Fun 𝐹𝐹 ⊆ (𝐴 × 𝐵)) → (𝐹 Fn dom 𝐹 ∧ ran 𝐹𝐵))
7 df-f 6565 . . . 4 (𝐹:dom 𝐹𝐵 ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹𝐵))
86, 7sylibr 234 . . 3 ((Fun 𝐹𝐹 ⊆ (𝐴 × 𝐵)) → 𝐹:dom 𝐹𝐵)
9 dmss 5913 . . . . 5 (𝐹 ⊆ (𝐴 × 𝐵) → dom 𝐹 ⊆ dom (𝐴 × 𝐵))
10 dmxpss 6191 . . . . 5 dom (𝐴 × 𝐵) ⊆ 𝐴
119, 10sstrdi 3996 . . . 4 (𝐹 ⊆ (𝐴 × 𝐵) → dom 𝐹𝐴)
1211adantl 481 . . 3 ((Fun 𝐹𝐹 ⊆ (𝐴 × 𝐵)) → dom 𝐹𝐴)
138, 12jca 511 . 2 ((Fun 𝐹𝐹 ⊆ (𝐴 × 𝐵)) → (𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴))
14 ffun 6739 . . . 4 (𝐹:dom 𝐹𝐵 → Fun 𝐹)
1514adantr 480 . . 3 ((𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴) → Fun 𝐹)
16 fssxp 6763 . . . 4 (𝐹:dom 𝐹𝐵𝐹 ⊆ (dom 𝐹 × 𝐵))
17 xpss1 5704 . . . 4 (dom 𝐹𝐴 → (dom 𝐹 × 𝐵) ⊆ (𝐴 × 𝐵))
1816, 17sylan9ss 3997 . . 3 ((𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴) → 𝐹 ⊆ (𝐴 × 𝐵))
1915, 18jca 511 . 2 ((𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴) → (Fun 𝐹𝐹 ⊆ (𝐴 × 𝐵)))
2013, 19impbii 209 1 ((Fun 𝐹𝐹 ⊆ (𝐴 × 𝐵)) ↔ (𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wss 3951   × cxp 5683  dom cdm 5685  ran crn 5686  Fun wfun 6555   Fn wfn 6556  wf 6557
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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-fun 6563  df-fn 6564  df-f 6565
This theorem is referenced by:  elpm2g  8884  volf  25564  dfno2  43441
  Copyright terms: Public domain W3C validator