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Theorem funssxp 5060
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 4931 . . . . . 6 (Fun 𝐹𝐹 Fn dom 𝐹)
21biimpi 113 . . . . 5 (Fun 𝐹𝐹 Fn dom 𝐹)
3 rnss 4564 . . . . . 6 (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹 ⊆ ran (𝐴 × 𝐵))
4 rnxpss 4754 . . . . . 6 ran (𝐴 × 𝐵) ⊆ 𝐵
53, 4syl6ss 2957 . . . . 5 (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹𝐵)
62, 5anim12i 321 . . . 4 ((Fun 𝐹𝐹 ⊆ (𝐴 × 𝐵)) → (𝐹 Fn dom 𝐹 ∧ ran 𝐹𝐵))
7 df-f 4906 . . . 4 (𝐹:dom 𝐹𝐵 ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹𝐵))
86, 7sylibr 137 . . 3 ((Fun 𝐹𝐹 ⊆ (𝐴 × 𝐵)) → 𝐹:dom 𝐹𝐵)
9 dmss 4534 . . . . 5 (𝐹 ⊆ (𝐴 × 𝐵) → dom 𝐹 ⊆ dom (𝐴 × 𝐵))
10 dmxpss 4753 . . . . 5 dom (𝐴 × 𝐵) ⊆ 𝐴
119, 10syl6ss 2957 . . . 4 (𝐹 ⊆ (𝐴 × 𝐵) → dom 𝐹𝐴)
1211adantl 262 . . 3 ((Fun 𝐹𝐹 ⊆ (𝐴 × 𝐵)) → dom 𝐹𝐴)
138, 12jca 290 . 2 ((Fun 𝐹𝐹 ⊆ (𝐴 × 𝐵)) → (𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴))
14 ffun 5048 . . . 4 (𝐹:dom 𝐹𝐵 → Fun 𝐹)
1514adantr 261 . . 3 ((𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴) → Fun 𝐹)
16 fssxp 5058 . . . 4 (𝐹:dom 𝐹𝐵𝐹 ⊆ (dom 𝐹 × 𝐵))
17 xpss1 4448 . . . 4 (dom 𝐹𝐴 → (dom 𝐹 × 𝐵) ⊆ (𝐴 × 𝐵))
1816, 17sylan9ss 2958 . . 3 ((𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴) → 𝐹 ⊆ (𝐴 × 𝐵))
1915, 18jca 290 . 2 ((𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴) → (Fun 𝐹𝐹 ⊆ (𝐴 × 𝐵)))
2013, 19impbii 117 1 ((Fun 𝐹𝐹 ⊆ (𝐴 × 𝐵)) ↔ (𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴))
Colors of variables: wff set class
Syntax hints:  wa 97  wb 98  wss 2917   × cxp 4343  dom cdm 4345  ran crn 4346  Fun wfun 4896   Fn wfn 4897  wf 4898
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356  df-fun 4904  df-fn 4905  df-f 4906
This theorem is referenced by: (None)
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