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Theorem ffvresb 5431
 Description: A necessary and sufficient condition for a restricted function. (Contributed by Mario Carneiro, 14-Nov-2013.)
Assertion
Ref Expression
ffvresb (Fun F → ((F A):A–→Bx A (x dom F (Fx) B)))
Distinct variable groups:   x,A   x,B   x,F

Proof of Theorem ffvresb
StepHypRef Expression
1 fdm 5226 . . . . . 6 ((F A):A–→B → dom (F A) = A)
2 dmres 4986 . . . . . . . 8 dom (F A) = (A ∩ dom F)
3 inss2 3476 . . . . . . . 8 (A ∩ dom F) dom F
42, 3eqsstri 3301 . . . . . . 7 dom (F A) dom F
54a1i 10 . . . . . 6 ((F A):A–→B → dom (F A) dom F)
61, 5eqsstr3d 3306 . . . . 5 ((F A):A–→BA dom F)
76sselda 3273 . . . 4 (((F A):A–→B x A) → x dom F)
8 fvres 5342 . . . . . 6 (x A → ((F A) ‘x) = (Fx))
98adantl 452 . . . . 5 (((F A):A–→B x A) → ((F A) ‘x) = (Fx))
10 ffvelrn 5415 . . . . 5 (((F A):A–→B x A) → ((F A) ‘x) B)
119, 10eqeltrrd 2428 . . . 4 (((F A):A–→B x A) → (Fx) B)
127, 11jca 518 . . 3 (((F A):A–→B x A) → (x dom F (Fx) B))
1312ralrimiva 2697 . 2 ((F A):A–→Bx A (x dom F (Fx) B))
14 simpl 443 . . . . . . 7 ((x dom F (Fx) B) → x dom F)
1514ralimi 2689 . . . . . 6 (x A (x dom F (Fx) B) → x A x dom F)
16 dfss3 3263 . . . . . 6 (A dom Fx A x dom F)
1715, 16sylibr 203 . . . . 5 (x A (x dom F (Fx) B) → A dom F)
18 funfn 5136 . . . . . 6 (Fun FF Fn dom F)
19 fnssres 5196 . . . . . 6 ((F Fn dom F A dom F) → (F A) Fn A)
2018, 19sylanb 458 . . . . 5 ((Fun F A dom F) → (F A) Fn A)
2117, 20sylan2 460 . . . 4 ((Fun F x A (x dom F (Fx) B)) → (F A) Fn A)
22 simpr 447 . . . . . . . 8 ((x dom F (Fx) B) → (Fx) B)
238eleq1d 2419 . . . . . . . 8 (x A → (((F A) ‘x) B ↔ (Fx) B))
2422, 23syl5ibr 212 . . . . . . 7 (x A → ((x dom F (Fx) B) → ((F A) ‘x) B))
2524ralimia 2687 . . . . . 6 (x A (x dom F (Fx) B) → x A ((F A) ‘x) B)
2625adantl 452 . . . . 5 ((Fun F x A (x dom F (Fx) B)) → x A ((F A) ‘x) B)
27 fnfvrnss 5429 . . . . 5 (((F A) Fn A x A ((F A) ‘x) B) → ran (F A) B)
2821, 26, 27syl2anc 642 . . . 4 ((Fun F x A (x dom F (Fx) B)) → ran (F A) B)
29 df-f 4791 . . . 4 ((F A):A–→B ↔ ((F A) Fn A ran (F A) B))
3021, 28, 29sylanbrc 645 . . 3 ((Fun F x A (x dom F (Fx) B)) → (F A):A–→B)
3130ex 423 . 2 (Fun F → (x A (x dom F (Fx) B) → (F A):A–→B))
3213, 31impbid2 195 1 (Fun F → ((F A):A–→Bx A (x dom F (Fx) B)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2614   ∩ cin 3208   ⊆ wss 3257  dom cdm 4772  ran crn 4773   ↾ cres 4774  Fun wfun 4775   Fn wfn 4776  –→wf 4777   ‘cfv 4781 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-fv 4795 This theorem is referenced by: (None)
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