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Theorem dff2 6682
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 6338 . . 3 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 fssxp 6357 . . 3 (𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
31, 2jca 504 . 2 (𝐹:𝐴𝐵 → (𝐹 Fn 𝐴𝐹 ⊆ (𝐴 × 𝐵)))
4 rnss 5645 . . . . 5 (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹 ⊆ ran (𝐴 × 𝐵))
5 rnxpss 5863 . . . . 5 ran (𝐴 × 𝐵) ⊆ 𝐵
64, 5syl6ss 3866 . . . 4 (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹𝐵)
76anim2i 607 . . 3 ((𝐹 Fn 𝐴𝐹 ⊆ (𝐴 × 𝐵)) → (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
8 df-f 6186 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
97, 8sylibr 226 . 2 ((𝐹 Fn 𝐴𝐹 ⊆ (𝐴 × 𝐵)) → 𝐹:𝐴𝐵)
103, 9impbii 201 1 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴𝐹 ⊆ (𝐴 × 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 387  wss 3825   × cxp 5398  ran crn 5401   Fn wfn 6177  wf 6178
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pr 5180
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-br 4924  df-opab 4986  df-xp 5406  df-rel 5407  df-cnv 5408  df-dm 5410  df-rn 5411  df-fun 6184  df-fn 6185  df-f 6186
This theorem is referenced by:  fpr2g  6794  mapval2  8228  cardf2  9158  imasaddflem  16649  imasvscaf  16658
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