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Theorem ffvresb 5271
Description: A necessary and sufficient condition for a restricted function. (Contributed by Mario Carneiro, 14-Nov-2013.)
Assertion
Ref Expression
ffvresb (Fun 𝐹 → ((𝐹A):ABx A (x dom 𝐹 (𝐹x) B)))
Distinct variable groups:   x,A   x,B   x,𝐹

Proof of Theorem ffvresb
StepHypRef Expression
1 fdm 4993 . . . . . 6 ((𝐹A):AB → dom (𝐹A) = A)
2 dmres 4575 . . . . . . 7 dom (𝐹A) = (A ∩ dom 𝐹)
3 inss2 3152 . . . . . . 7 (A ∩ dom 𝐹) ⊆ dom 𝐹
42, 3eqsstri 2969 . . . . . 6 dom (𝐹A) ⊆ dom 𝐹
51, 4syl6eqssr 2990 . . . . 5 ((𝐹A):ABA ⊆ dom 𝐹)
65sselda 2939 . . . 4 (((𝐹A):AB x A) → x dom 𝐹)
7 fvres 5141 . . . . . 6 (x A → ((𝐹A)‘x) = (𝐹x))
87adantl 262 . . . . 5 (((𝐹A):AB x A) → ((𝐹A)‘x) = (𝐹x))
9 ffvelrn 5243 . . . . 5 (((𝐹A):AB x A) → ((𝐹A)‘x) B)
108, 9eqeltrrd 2112 . . . 4 (((𝐹A):AB x A) → (𝐹x) B)
116, 10jca 290 . . 3 (((𝐹A):AB x A) → (x dom 𝐹 (𝐹x) B))
1211ralrimiva 2386 . 2 ((𝐹A):ABx A (x dom 𝐹 (𝐹x) B))
13 simpl 102 . . . . . . 7 ((x dom 𝐹 (𝐹x) B) → x dom 𝐹)
1413ralimi 2378 . . . . . 6 (x A (x dom 𝐹 (𝐹x) B) → x A x dom 𝐹)
15 dfss3 2929 . . . . . 6 (A ⊆ dom 𝐹x A x dom 𝐹)
1614, 15sylibr 137 . . . . 5 (x A (x dom 𝐹 (𝐹x) B) → A ⊆ dom 𝐹)
17 funfn 4874 . . . . . 6 (Fun 𝐹𝐹 Fn dom 𝐹)
18 fnssres 4955 . . . . . 6 ((𝐹 Fn dom 𝐹 A ⊆ dom 𝐹) → (𝐹A) Fn A)
1917, 18sylanb 268 . . . . 5 ((Fun 𝐹 A ⊆ dom 𝐹) → (𝐹A) Fn A)
2016, 19sylan2 270 . . . 4 ((Fun 𝐹 x A (x dom 𝐹 (𝐹x) B)) → (𝐹A) Fn A)
21 simpr 103 . . . . . . . 8 ((x dom 𝐹 (𝐹x) B) → (𝐹x) B)
227eleq1d 2103 . . . . . . . 8 (x A → (((𝐹A)‘x) B ↔ (𝐹x) B))
2321, 22syl5ibr 145 . . . . . . 7 (x A → ((x dom 𝐹 (𝐹x) B) → ((𝐹A)‘x) B))
2423ralimia 2376 . . . . . 6 (x A (x dom 𝐹 (𝐹x) B) → x A ((𝐹A)‘x) B)
2524adantl 262 . . . . 5 ((Fun 𝐹 x A (x dom 𝐹 (𝐹x) B)) → x A ((𝐹A)‘x) B)
26 fnfvrnss 5268 . . . . 5 (((𝐹A) Fn A x A ((𝐹A)‘x) B) → ran (𝐹A) ⊆ B)
2720, 25, 26syl2anc 391 . . . 4 ((Fun 𝐹 x A (x dom 𝐹 (𝐹x) B)) → ran (𝐹A) ⊆ B)
28 df-f 4849 . . . 4 ((𝐹A):AB ↔ ((𝐹A) Fn A ran (𝐹A) ⊆ B))
2920, 27, 28sylanbrc 394 . . 3 ((Fun 𝐹 x A (x dom 𝐹 (𝐹x) B)) → (𝐹A):AB)
3029ex 108 . 2 (Fun 𝐹 → (x A (x dom 𝐹 (𝐹x) B) → (𝐹A):AB))
3112, 30impbid2 131 1 (Fun 𝐹 → ((𝐹A):ABx A (x dom 𝐹 (𝐹x) B)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  wral 2300  cin 2910  wss 2911  dom cdm 4288  ran crn 4289  cres 4290  Fun wfun 4839   Fn wfn 4840  wf 4841  cfv 4845
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-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fv 4853
This theorem is referenced by: (None)
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