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Theorem dff2 7119
Description: Alternate definition of a mapping. (Contributed by NM, 14-Nov-2007.)
Assertion
Ref Expression
dff2 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴𝐹 ⊆ (𝐴 × 𝐵)))

Proof of Theorem dff2
StepHypRef Expression
1 ffn 6737 . . 3 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 fssxp 6764 . . 3 (𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
31, 2jca 511 . 2 (𝐹:𝐴𝐵 → (𝐹 Fn 𝐴𝐹 ⊆ (𝐴 × 𝐵)))
4 rnss 5953 . . . . 5 (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹 ⊆ ran (𝐴 × 𝐵))
5 rnxpss 6194 . . . . 5 ran (𝐴 × 𝐵) ⊆ 𝐵
64, 5sstrdi 4008 . . . 4 (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹𝐵)
76anim2i 617 . . 3 ((𝐹 Fn 𝐴𝐹 ⊆ (𝐴 × 𝐵)) → (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
8 df-f 6567 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
97, 8sylibr 234 . 2 ((𝐹 Fn 𝐴𝐹 ⊆ (𝐴 × 𝐵)) → 𝐹:𝐴𝐵)
103, 9impbii 209 1 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴𝐹 ⊆ (𝐴 × 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wss 3963   × cxp 5687  ran crn 5690   Fn wfn 6558  wf 6559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-fun 6565  df-fn 6566  df-f 6567
This theorem is referenced by:  fpr2g  7231  mapval2  8911  cardf2  9981  mpoaddf  11247  mpomulf  11248  imasaddflem  17577  imasvscaf  17586  gsumpart  33043
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