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Theorem dff2 6279
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 5958 . . 3 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 fssxp 5973 . . 3 (𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
31, 2jca 553 . 2 (𝐹:𝐴𝐵 → (𝐹 Fn 𝐴𝐹 ⊆ (𝐴 × 𝐵)))
4 rnss 5275 . . . . 5 (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹 ⊆ ran (𝐴 × 𝐵))
5 rnxpss 5485 . . . . 5 ran (𝐴 × 𝐵) ⊆ 𝐵
64, 5syl6ss 3580 . . . 4 (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹𝐵)
76anim2i 591 . . 3 ((𝐹 Fn 𝐴𝐹 ⊆ (𝐴 × 𝐵)) → (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
8 df-f 5808 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
97, 8sylibr 223 . 2 ((𝐹 Fn 𝐴𝐹 ⊆ (𝐴 × 𝐵)) → 𝐹:𝐴𝐵)
103, 9impbii 198 1 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴𝐹 ⊆ (𝐴 × 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wa 383  wss 3540   × cxp 5036  ran crn 5039   Fn wfn 5799  wf 5800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-fun 5806  df-fn 5807  df-f 5808
This theorem is referenced by:  fpr2g  6380  mapval2  7773  cardf2  8652  imasaddflem  16013  imasvscaf  16022
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