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Theorem funssxp 5957
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 5816 . . . . . 6 (Fun 𝐹𝐹 Fn dom 𝐹)
21biimpi 204 . . . . 5 (Fun 𝐹𝐹 Fn dom 𝐹)
3 rnss 5259 . . . . . 6 (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹 ⊆ ran (𝐴 × 𝐵))
4 rnxpss 5468 . . . . . 6 ran (𝐴 × 𝐵) ⊆ 𝐵
53, 4syl6ss 3576 . . . . 5 (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹𝐵)
62, 5anim12i 587 . . . 4 ((Fun 𝐹𝐹 ⊆ (𝐴 × 𝐵)) → (𝐹 Fn dom 𝐹 ∧ ran 𝐹𝐵))
7 df-f 5791 . . . 4 (𝐹:dom 𝐹𝐵 ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹𝐵))
86, 7sylibr 222 . . 3 ((Fun 𝐹𝐹 ⊆ (𝐴 × 𝐵)) → 𝐹:dom 𝐹𝐵)
9 dmss 5229 . . . . 5 (𝐹 ⊆ (𝐴 × 𝐵) → dom 𝐹 ⊆ dom (𝐴 × 𝐵))
10 dmxpss 5467 . . . . 5 dom (𝐴 × 𝐵) ⊆ 𝐴
119, 10syl6ss 3576 . . . 4 (𝐹 ⊆ (𝐴 × 𝐵) → dom 𝐹𝐴)
1211adantl 480 . . 3 ((Fun 𝐹𝐹 ⊆ (𝐴 × 𝐵)) → dom 𝐹𝐴)
138, 12jca 552 . 2 ((Fun 𝐹𝐹 ⊆ (𝐴 × 𝐵)) → (𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴))
14 ffun 5944 . . . 4 (𝐹:dom 𝐹𝐵 → Fun 𝐹)
1514adantr 479 . . 3 ((𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴) → Fun 𝐹)
16 fssxp 5956 . . . 4 (𝐹:dom 𝐹𝐵𝐹 ⊆ (dom 𝐹 × 𝐵))
17 xpss1 5137 . . . 4 (dom 𝐹𝐴 → (dom 𝐹 × 𝐵) ⊆ (𝐴 × 𝐵))
1816, 17sylan9ss 3577 . . 3 ((𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴) → 𝐹 ⊆ (𝐴 × 𝐵))
1915, 18jca 552 . 2 ((𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴) → (Fun 𝐹𝐹 ⊆ (𝐴 × 𝐵)))
2013, 19impbii 197 1 ((Fun 𝐹𝐹 ⊆ (𝐴 × 𝐵)) ↔ (𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382  wss 3536   × cxp 5023  dom cdm 5025  ran crn 5026  Fun wfun 5781   Fn wfn 5782  wf 5783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pr 4825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-rab 2901  df-v 3171  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-sn 4122  df-pr 4124  df-op 4128  df-br 4575  df-opab 4635  df-xp 5031  df-rel 5032  df-cnv 5033  df-dm 5035  df-rn 5036  df-fun 5789  df-fn 5790  df-f 5791
This theorem is referenced by:  elpm2g  7734  volf  23018
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