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Theorem funssxp 6746
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 6578 . . . . . 6 (Fun 𝐹𝐹 Fn dom 𝐹)
21biimpi 215 . . . . 5 (Fun 𝐹𝐹 Fn dom 𝐹)
3 rnss 5938 . . . . . 6 (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹 ⊆ ran (𝐴 × 𝐵))
4 rnxpss 6171 . . . . . 6 ran (𝐴 × 𝐵) ⊆ 𝐵
53, 4sstrdi 3994 . . . . 5 (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹𝐵)
62, 5anim12i 612 . . . 4 ((Fun 𝐹𝐹 ⊆ (𝐴 × 𝐵)) → (𝐹 Fn dom 𝐹 ∧ ran 𝐹𝐵))
7 df-f 6547 . . . 4 (𝐹:dom 𝐹𝐵 ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹𝐵))
86, 7sylibr 233 . . 3 ((Fun 𝐹𝐹 ⊆ (𝐴 × 𝐵)) → 𝐹:dom 𝐹𝐵)
9 dmss 5902 . . . . 5 (𝐹 ⊆ (𝐴 × 𝐵) → dom 𝐹 ⊆ dom (𝐴 × 𝐵))
10 dmxpss 6170 . . . . 5 dom (𝐴 × 𝐵) ⊆ 𝐴
119, 10sstrdi 3994 . . . 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 6720 . . . 4 (𝐹:dom 𝐹𝐵 → Fun 𝐹)
1514adantr 480 . . 3 ((𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴) → Fun 𝐹)
16 fssxp 6745 . . . 4 (𝐹:dom 𝐹𝐵𝐹 ⊆ (dom 𝐹 × 𝐵))
17 xpss1 5695 . . . 4 (dom 𝐹𝐴 → (dom 𝐹 × 𝐵) ⊆ (𝐴 × 𝐵))
1816, 17sylan9ss 3995 . . 3 ((𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴) → 𝐹 ⊆ (𝐴 × 𝐵))
1915, 18jca 511 . 2 ((𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴) → (Fun 𝐹𝐹 ⊆ (𝐴 × 𝐵)))
2013, 19impbii 208 1 ((Fun 𝐹𝐹 ⊆ (𝐴 × 𝐵)) ↔ (𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  wss 3948   × cxp 5674  dom cdm 5676  ran crn 5677  Fun wfun 6537   Fn wfn 6538  wf 6539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-dm 5686  df-rn 5687  df-fun 6545  df-fn 6546  df-f 6547
This theorem is referenced by:  elpm2g  8844  volf  25378  dfno2  42642
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