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Theorem pmresg 7829
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
StepHypRef Expression
1 n0i 3896 . . . . 5 (𝐹 ∈ (𝐴pm 𝐶) → ¬ (𝐴pm 𝐶) = ∅)
2 fnpm 7810 . . . . . . 7 pm Fn (V × V)
3 fndm 5948 . . . . . . 7 ( ↑pm Fn (V × V) → dom ↑pm = (V × V))
42, 3ax-mp 5 . . . . . 6 dom ↑pm = (V × V)
54ndmov 6771 . . . . 5 (¬ (𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴pm 𝐶) = ∅)
61, 5nsyl2 142 . . . 4 (𝐹 ∈ (𝐴pm 𝐶) → (𝐴 ∈ V ∧ 𝐶 ∈ V))
76simpld 475 . . 3 (𝐹 ∈ (𝐴pm 𝐶) → 𝐴 ∈ V)
87adantl 482 . 2 ((𝐵𝑉𝐹 ∈ (𝐴pm 𝐶)) → 𝐴 ∈ V)
9 simpl 473 . 2 ((𝐵𝑉𝐹 ∈ (𝐴pm 𝐶)) → 𝐵𝑉)
10 elpmi 7820 . . . . . 6 (𝐹 ∈ (𝐴pm 𝐶) → (𝐹:dom 𝐹𝐴 ∧ dom 𝐹𝐶))
1110simpld 475 . . . . 5 (𝐹 ∈ (𝐴pm 𝐶) → 𝐹:dom 𝐹𝐴)
1211adantl 482 . . . 4 ((𝐵𝑉𝐹 ∈ (𝐴pm 𝐶)) → 𝐹:dom 𝐹𝐴)
13 inss1 3811 . . . 4 (dom 𝐹𝐵) ⊆ dom 𝐹
14 fssres 6027 . . . 4 ((𝐹:dom 𝐹𝐴 ∧ (dom 𝐹𝐵) ⊆ dom 𝐹) → (𝐹 ↾ (dom 𝐹𝐵)):(dom 𝐹𝐵)⟶𝐴)
1512, 13, 14sylancl 693 . . 3 ((𝐵𝑉𝐹 ∈ (𝐴pm 𝐶)) → (𝐹 ↾ (dom 𝐹𝐵)):(dom 𝐹𝐵)⟶𝐴)
16 ffun 6005 . . . . 5 (𝐹:dom 𝐹𝐴 → Fun 𝐹)
17 resres 5368 . . . . . 6 ((𝐹 ↾ dom 𝐹) ↾ 𝐵) = (𝐹 ↾ (dom 𝐹𝐵))
18 funrel 5864 . . . . . . 7 (Fun 𝐹 → Rel 𝐹)
19 resdm 5400 . . . . . . 7 (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
20 reseq1 5350 . . . . . . 7 ((𝐹 ↾ dom 𝐹) = 𝐹 → ((𝐹 ↾ dom 𝐹) ↾ 𝐵) = (𝐹𝐵))
2118, 19, 203syl 18 . . . . . 6 (Fun 𝐹 → ((𝐹 ↾ dom 𝐹) ↾ 𝐵) = (𝐹𝐵))
2217, 21syl5eqr 2669 . . . . 5 (Fun 𝐹 → (𝐹 ↾ (dom 𝐹𝐵)) = (𝐹𝐵))
2312, 16, 223syl 18 . . . 4 ((𝐵𝑉𝐹 ∈ (𝐴pm 𝐶)) → (𝐹 ↾ (dom 𝐹𝐵)) = (𝐹𝐵))
2423feq1d 5987 . . 3 ((𝐵𝑉𝐹 ∈ (𝐴pm 𝐶)) → ((𝐹 ↾ (dom 𝐹𝐵)):(dom 𝐹𝐵)⟶𝐴 ↔ (𝐹𝐵):(dom 𝐹𝐵)⟶𝐴))
2515, 24mpbid 222 . 2 ((𝐵𝑉𝐹 ∈ (𝐴pm 𝐶)) → (𝐹𝐵):(dom 𝐹𝐵)⟶𝐴)
26 inss2 3812 . . 3 (dom 𝐹𝐵) ⊆ 𝐵
27 elpm2r 7819 . . 3 (((𝐴 ∈ V ∧ 𝐵𝑉) ∧ ((𝐹𝐵):(dom 𝐹𝐵)⟶𝐴 ∧ (dom 𝐹𝐵) ⊆ 𝐵)) → (𝐹𝐵) ∈ (𝐴pm 𝐵))
2826, 27mpanr2 719 . 2 (((𝐴 ∈ V ∧ 𝐵𝑉) ∧ (𝐹𝐵):(dom 𝐹𝐵)⟶𝐴) → (𝐹𝐵) ∈ (𝐴pm 𝐵))
298, 9, 25, 28syl21anc 1322 1 ((𝐵𝑉𝐹 ∈ (𝐴pm 𝐶)) → (𝐹𝐵) ∈ (𝐴pm 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  cin 3554  wss 3555  c0 3891   × cxp 5072  dom cdm 5074  cres 5076  Rel wrel 5079  Fun wfun 5841   Fn wfn 5842  wf 5843  (class class class)co 6604  pm cpm 7803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-pm 7805
This theorem is referenced by:  lmres  21014  mbfres  23317  dvnres  23600  cpnres  23606  caures  33188
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