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Theorem funssxp 6757
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 6588 . . . . . 6 (Fun 𝐹𝐹 Fn dom 𝐹)
21biimpi 215 . . . . 5 (Fun 𝐹𝐹 Fn dom 𝐹)
3 rnss 5945 . . . . . 6 (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹 ⊆ ran (𝐴 × 𝐵))
4 rnxpss 6181 . . . . . 6 ran (𝐴 × 𝐵) ⊆ 𝐵
53, 4sstrdi 3994 . . . . 5 (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹𝐵)
62, 5anim12i 611 . . . 4 ((Fun 𝐹𝐹 ⊆ (𝐴 × 𝐵)) → (𝐹 Fn dom 𝐹 ∧ ran 𝐹𝐵))
7 df-f 6557 . . . 4 (𝐹:dom 𝐹𝐵 ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹𝐵))
86, 7sylibr 233 . . 3 ((Fun 𝐹𝐹 ⊆ (𝐴 × 𝐵)) → 𝐹:dom 𝐹𝐵)
9 dmss 5909 . . . . 5 (𝐹 ⊆ (𝐴 × 𝐵) → dom 𝐹 ⊆ dom (𝐴 × 𝐵))
10 dmxpss 6180 . . . . 5 dom (𝐴 × 𝐵) ⊆ 𝐴
119, 10sstrdi 3994 . . . 4 (𝐹 ⊆ (𝐴 × 𝐵) → dom 𝐹𝐴)
1211adantl 480 . . 3 ((Fun 𝐹𝐹 ⊆ (𝐴 × 𝐵)) → dom 𝐹𝐴)
138, 12jca 510 . 2 ((Fun 𝐹𝐹 ⊆ (𝐴 × 𝐵)) → (𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴))
14 ffun 6730 . . . 4 (𝐹:dom 𝐹𝐵 → Fun 𝐹)
1514adantr 479 . . 3 ((𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴) → Fun 𝐹)
16 fssxp 6756 . . . 4 (𝐹:dom 𝐹𝐵𝐹 ⊆ (dom 𝐹 × 𝐵))
17 xpss1 5701 . . . 4 (dom 𝐹𝐴 → (dom 𝐹 × 𝐵) ⊆ (𝐴 × 𝐵))
1816, 17sylan9ss 3995 . . 3 ((𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴) → 𝐹 ⊆ (𝐴 × 𝐵))
1915, 18jca 510 . 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 394  wss 3949   × cxp 5680  dom cdm 5682  ran crn 5683  Fun wfun 6547   Fn wfn 6548  wf 6549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-xp 5688  df-rel 5689  df-cnv 5690  df-dm 5692  df-rn 5693  df-fun 6555  df-fn 6556  df-f 6557
This theorem is referenced by:  elpm2g  8869  volf  25478  dfno2  42889
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