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Theorem pmresg 6676
Description: Elementhood of a restricted function in the set of partial functions. (Contributed by Mario Carneiro, 31-Dec-2013.)
Assertion
Ref Expression
pmresg ((𝐵𝑉𝐹 ∈ (𝐴pm 𝐶)) → (𝐹𝐵) ∈ (𝐴pm 𝐵))

Proof of Theorem pmresg
Dummy variables 𝑥 𝑓 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-pm 6651 . . . 4 pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓})
21elmpocl1 6070 . . 3 (𝐹 ∈ (𝐴pm 𝐶) → 𝐴 ∈ V)
32adantl 277 . 2 ((𝐵𝑉𝐹 ∈ (𝐴pm 𝐶)) → 𝐴 ∈ V)
4 simpl 109 . 2 ((𝐵𝑉𝐹 ∈ (𝐴pm 𝐶)) → 𝐵𝑉)
5 elpmi 6667 . . . . . 6 (𝐹 ∈ (𝐴pm 𝐶) → (𝐹:dom 𝐹𝐴 ∧ dom 𝐹𝐶))
65simpld 112 . . . . 5 (𝐹 ∈ (𝐴pm 𝐶) → 𝐹:dom 𝐹𝐴)
76adantl 277 . . . 4 ((𝐵𝑉𝐹 ∈ (𝐴pm 𝐶)) → 𝐹:dom 𝐹𝐴)
8 inss1 3356 . . . 4 (dom 𝐹𝐵) ⊆ dom 𝐹
9 fssres 5392 . . . 4 ((𝐹:dom 𝐹𝐴 ∧ (dom 𝐹𝐵) ⊆ dom 𝐹) → (𝐹 ↾ (dom 𝐹𝐵)):(dom 𝐹𝐵)⟶𝐴)
107, 8, 9sylancl 413 . . 3 ((𝐵𝑉𝐹 ∈ (𝐴pm 𝐶)) → (𝐹 ↾ (dom 𝐹𝐵)):(dom 𝐹𝐵)⟶𝐴)
11 ffun 5369 . . . . 5 (𝐹:dom 𝐹𝐴 → Fun 𝐹)
12 resres 4920 . . . . . 6 ((𝐹 ↾ dom 𝐹) ↾ 𝐵) = (𝐹 ↾ (dom 𝐹𝐵))
13 funrel 5234 . . . . . . 7 (Fun 𝐹 → Rel 𝐹)
14 resdm 4947 . . . . . . 7 (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
15 reseq1 4902 . . . . . . 7 ((𝐹 ↾ dom 𝐹) = 𝐹 → ((𝐹 ↾ dom 𝐹) ↾ 𝐵) = (𝐹𝐵))
1613, 14, 153syl 17 . . . . . 6 (Fun 𝐹 → ((𝐹 ↾ dom 𝐹) ↾ 𝐵) = (𝐹𝐵))
1712, 16eqtr3id 2224 . . . . 5 (Fun 𝐹 → (𝐹 ↾ (dom 𝐹𝐵)) = (𝐹𝐵))
187, 11, 173syl 17 . . . 4 ((𝐵𝑉𝐹 ∈ (𝐴pm 𝐶)) → (𝐹 ↾ (dom 𝐹𝐵)) = (𝐹𝐵))
1918feq1d 5353 . . 3 ((𝐵𝑉𝐹 ∈ (𝐴pm 𝐶)) → ((𝐹 ↾ (dom 𝐹𝐵)):(dom 𝐹𝐵)⟶𝐴 ↔ (𝐹𝐵):(dom 𝐹𝐵)⟶𝐴))
2010, 19mpbid 147 . 2 ((𝐵𝑉𝐹 ∈ (𝐴pm 𝐶)) → (𝐹𝐵):(dom 𝐹𝐵)⟶𝐴)
21 inss2 3357 . . 3 (dom 𝐹𝐵) ⊆ 𝐵
22 elpm2r 6666 . . 3 (((𝐴 ∈ V ∧ 𝐵𝑉) ∧ ((𝐹𝐵):(dom 𝐹𝐵)⟶𝐴 ∧ (dom 𝐹𝐵) ⊆ 𝐵)) → (𝐹𝐵) ∈ (𝐴pm 𝐵))
2321, 22mpanr2 438 . 2 (((𝐴 ∈ V ∧ 𝐵𝑉) ∧ (𝐹𝐵):(dom 𝐹𝐵)⟶𝐴) → (𝐹𝐵) ∈ (𝐴pm 𝐵))
243, 4, 20, 23syl21anc 1237 1 ((𝐵𝑉𝐹 ∈ (𝐴pm 𝐶)) → (𝐹𝐵) ∈ (𝐴pm 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  {crab 2459  Vcvv 2738  cin 3129  wss 3130  𝒫 cpw 3576   × cxp 4625  dom cdm 4627  cres 4629  Rel wrel 4632  Fun wfun 5211  wf 5213  (class class class)co 5875  pm cpm 6649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-pm 6651
This theorem is referenced by:  lmres  13751
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