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Theorem funssxp 5003
 Description: Two ways of specifying a partial function from A to B. (Contributed by NM, 13-Nov-2007.)
Assertion
Ref Expression
funssxp ((Fun 𝐹 𝐹 ⊆ (A × B)) ↔ (𝐹:dom 𝐹B dom 𝐹A))

Proof of Theorem funssxp
StepHypRef Expression
1 funfn 4874 . . . . . 6 (Fun 𝐹𝐹 Fn dom 𝐹)
21biimpi 113 . . . . 5 (Fun 𝐹𝐹 Fn dom 𝐹)
3 rnss 4507 . . . . . 6 (𝐹 ⊆ (A × B) → ran 𝐹 ⊆ ran (A × B))
4 rnxpss 4697 . . . . . 6 ran (A × B) ⊆ B
53, 4syl6ss 2951 . . . . 5 (𝐹 ⊆ (A × B) → ran 𝐹B)
62, 5anim12i 321 . . . 4 ((Fun 𝐹 𝐹 ⊆ (A × B)) → (𝐹 Fn dom 𝐹 ran 𝐹B))
7 df-f 4849 . . . 4 (𝐹:dom 𝐹B ↔ (𝐹 Fn dom 𝐹 ran 𝐹B))
86, 7sylibr 137 . . 3 ((Fun 𝐹 𝐹 ⊆ (A × B)) → 𝐹:dom 𝐹B)
9 dmss 4477 . . . . 5 (𝐹 ⊆ (A × B) → dom 𝐹 ⊆ dom (A × B))
10 dmxpss 4696 . . . . 5 dom (A × B) ⊆ A
119, 10syl6ss 2951 . . . 4 (𝐹 ⊆ (A × B) → dom 𝐹A)
1211adantl 262 . . 3 ((Fun 𝐹 𝐹 ⊆ (A × B)) → dom 𝐹A)
138, 12jca 290 . 2 ((Fun 𝐹 𝐹 ⊆ (A × B)) → (𝐹:dom 𝐹B dom 𝐹A))
14 ffun 4991 . . . 4 (𝐹:dom 𝐹B → Fun 𝐹)
1514adantr 261 . . 3 ((𝐹:dom 𝐹B dom 𝐹A) → Fun 𝐹)
16 fssxp 5001 . . . 4 (𝐹:dom 𝐹B𝐹 ⊆ (dom 𝐹 × B))
17 xpss1 4391 . . . 4 (dom 𝐹A → (dom 𝐹 × B) ⊆ (A × B))
1816, 17sylan9ss 2952 . . 3 ((𝐹:dom 𝐹B dom 𝐹A) → 𝐹 ⊆ (A × B))
1915, 18jca 290 . 2 ((𝐹:dom 𝐹B dom 𝐹A) → (Fun 𝐹 𝐹 ⊆ (A × B)))
2013, 19impbii 117 1 ((Fun 𝐹 𝐹 ⊆ (A × B)) ↔ (𝐹:dom 𝐹B dom 𝐹A))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   ⊆ wss 2911   × cxp 4286  dom cdm 4288  ran crn 4289  Fun wfun 4839   Fn wfn 4840  ⟶wf 4841 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849 This theorem is referenced by: (None)
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