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Theorem dff2 6327
 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 6002 . . 3 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 fssxp 6017 . . 3 (𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
31, 2jca 554 . 2 (𝐹:𝐴𝐵 → (𝐹 Fn 𝐴𝐹 ⊆ (𝐴 × 𝐵)))
4 rnss 5314 . . . . 5 (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹 ⊆ ran (𝐴 × 𝐵))
5 rnxpss 5525 . . . . 5 ran (𝐴 × 𝐵) ⊆ 𝐵
64, 5syl6ss 3595 . . . 4 (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹𝐵)
76anim2i 592 . . 3 ((𝐹 Fn 𝐴𝐹 ⊆ (𝐴 × 𝐵)) → (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
8 df-f 5851 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
97, 8sylibr 224 . 2 ((𝐹 Fn 𝐴𝐹 ⊆ (𝐴 × 𝐵)) → 𝐹:𝐴𝐵)
103, 9impbii 199 1 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴𝐹 ⊆ (𝐴 × 𝐵)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 384   ⊆ wss 3555   × cxp 5072  ran crn 5075   Fn wfn 5842  ⟶wf 5843 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-xp 5080  df-rel 5081  df-cnv 5082  df-dm 5084  df-rn 5085  df-fun 5849  df-fn 5850  df-f 5851 This theorem is referenced by:  fpr2g  6429  mapval2  7831  cardf2  8713  imasaddflem  16111  imasvscaf  16120
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