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Theorem dff2 5254
Description: Alternate definition of a mapping. (Contributed by NM, 14-Nov-2007.)
Assertion
Ref Expression
dff2 (𝐹:AB ↔ (𝐹 Fn A 𝐹 ⊆ (A × B)))

Proof of Theorem dff2
StepHypRef Expression
1 ffn 4989 . . 3 (𝐹:AB𝐹 Fn A)
2 fssxp 5001 . . 3 (𝐹:AB𝐹 ⊆ (A × B))
31, 2jca 290 . 2 (𝐹:AB → (𝐹 Fn A 𝐹 ⊆ (A × B)))
4 rnss 4507 . . . . 5 (𝐹 ⊆ (A × B) → ran 𝐹 ⊆ ran (A × B))
5 rnxpss 4697 . . . . 5 ran (A × B) ⊆ B
64, 5syl6ss 2951 . . . 4 (𝐹 ⊆ (A × B) → ran 𝐹B)
76anim2i 324 . . 3 ((𝐹 Fn A 𝐹 ⊆ (A × B)) → (𝐹 Fn A ran 𝐹B))
8 df-f 4849 . . 3 (𝐹:AB ↔ (𝐹 Fn A ran 𝐹B))
97, 8sylibr 137 . 2 ((𝐹 Fn A 𝐹 ⊆ (A × B)) → 𝐹:AB)
103, 9impbii 117 1 (𝐹:AB ↔ (𝐹 Fn A 𝐹 ⊆ (A × B)))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98  wss 2911   × cxp 4286  ran crn 4289   Fn wfn 4840  wf 4841
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849
This theorem is referenced by: (None)
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