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Theorem dff2 6846
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 6491 . . 3 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 fssxp 6512 . . 3 (𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
31, 2jca 515 . 2 (𝐹:𝐴𝐵 → (𝐹 Fn 𝐴𝐹 ⊆ (𝐴 × 𝐵)))
4 rnss 5777 . . . . 5 (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹 ⊆ ran (𝐴 × 𝐵))
5 rnxpss 6000 . . . . 5 ran (𝐴 × 𝐵) ⊆ 𝐵
64, 5sstrdi 3930 . . . 4 (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹𝐵)
76anim2i 619 . . 3 ((𝐹 Fn 𝐴𝐹 ⊆ (𝐴 × 𝐵)) → (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
8 df-f 6332 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
97, 8sylibr 237 . 2 ((𝐹 Fn 𝐴𝐹 ⊆ (𝐴 × 𝐵)) → 𝐹:𝐴𝐵)
103, 9impbii 212 1 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴𝐹 ⊆ (𝐴 × 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wss 3884   × cxp 5521  ran crn 5524   Fn wfn 6323  wf 6324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-xp 5529  df-rel 5530  df-cnv 5531  df-dm 5533  df-rn 5534  df-fun 6330  df-fn 6331  df-f 6332
This theorem is referenced by:  fpr2g  6955  mapval2  8423  cardf2  9360  imasaddflem  16799  imasvscaf  16808  gsumpart  30744
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