ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  suppssof1 GIF version

Theorem suppssof1 5872
Description: Formula building theorem for support restrictions: vector operation with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.)
Hypotheses
Ref Expression
suppssof1.s (𝜑 → (𝐴 “ (V ∖ {𝑌})) ⊆ 𝐿)
suppssof1.o ((𝜑𝑣𝑅) → (𝑌𝑂𝑣) = 𝑍)
suppssof1.a (𝜑𝐴:𝐷𝑉)
suppssof1.b (𝜑𝐵:𝐷𝑅)
suppssof1.d (𝜑𝐷𝑊)
Assertion
Ref Expression
suppssof1 (𝜑 → ((𝐴𝑓 𝑂𝐵) “ (V ∖ {𝑍})) ⊆ 𝐿)
Distinct variable groups:   𝜑,𝑣   𝑣,𝐵   𝑣,𝑂   𝑣,𝑅   𝑣,𝑌   𝑣,𝑍
Allowed substitution hints:   𝐴(𝑣)   𝐷(𝑣)   𝐿(𝑣)   𝑉(𝑣)   𝑊(𝑣)

Proof of Theorem suppssof1
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 suppssof1.a . . . . . 6 (𝜑𝐴:𝐷𝑉)
2 ffn 5161 . . . . . 6 (𝐴:𝐷𝑉𝐴 Fn 𝐷)
31, 2syl 14 . . . . 5 (𝜑𝐴 Fn 𝐷)
4 suppssof1.b . . . . . 6 (𝜑𝐵:𝐷𝑅)
5 ffn 5161 . . . . . 6 (𝐵:𝐷𝑅𝐵 Fn 𝐷)
64, 5syl 14 . . . . 5 (𝜑𝐵 Fn 𝐷)
7 suppssof1.d . . . . 5 (𝜑𝐷𝑊)
8 inidm 3209 . . . . 5 (𝐷𝐷) = 𝐷
9 eqidd 2089 . . . . 5 ((𝜑𝑥𝐷) → (𝐴𝑥) = (𝐴𝑥))
10 eqidd 2089 . . . . 5 ((𝜑𝑥𝐷) → (𝐵𝑥) = (𝐵𝑥))
113, 6, 7, 7, 8, 9, 10offval 5863 . . . 4 (𝜑 → (𝐴𝑓 𝑂𝐵) = (𝑥𝐷 ↦ ((𝐴𝑥)𝑂(𝐵𝑥))))
1211cnveqd 4612 . . 3 (𝜑(𝐴𝑓 𝑂𝐵) = (𝑥𝐷 ↦ ((𝐴𝑥)𝑂(𝐵𝑥))))
1312imaeq1d 4773 . 2 (𝜑 → ((𝐴𝑓 𝑂𝐵) “ (V ∖ {𝑍})) = ((𝑥𝐷 ↦ ((𝐴𝑥)𝑂(𝐵𝑥))) “ (V ∖ {𝑍})))
141feqmptd 5357 . . . . . 6 (𝜑𝐴 = (𝑥𝐷 ↦ (𝐴𝑥)))
1514cnveqd 4612 . . . . 5 (𝜑𝐴 = (𝑥𝐷 ↦ (𝐴𝑥)))
1615imaeq1d 4773 . . . 4 (𝜑 → (𝐴 “ (V ∖ {𝑌})) = ((𝑥𝐷 ↦ (𝐴𝑥)) “ (V ∖ {𝑌})))
17 suppssof1.s . . . 4 (𝜑 → (𝐴 “ (V ∖ {𝑌})) ⊆ 𝐿)
1816, 17eqsstr3d 3061 . . 3 (𝜑 → ((𝑥𝐷 ↦ (𝐴𝑥)) “ (V ∖ {𝑌})) ⊆ 𝐿)
19 suppssof1.o . . 3 ((𝜑𝑣𝑅) → (𝑌𝑂𝑣) = 𝑍)
20 funfvex 5322 . . . . 5 ((Fun 𝐴𝑥 ∈ dom 𝐴) → (𝐴𝑥) ∈ V)
2120funfni 5114 . . . 4 ((𝐴 Fn 𝐷𝑥𝐷) → (𝐴𝑥) ∈ V)
223, 21sylan 277 . . 3 ((𝜑𝑥𝐷) → (𝐴𝑥) ∈ V)
234ffvelrnda 5434 . . 3 ((𝜑𝑥𝐷) → (𝐵𝑥) ∈ 𝑅)
2418, 19, 22, 23suppssov1 5853 . 2 (𝜑 → ((𝑥𝐷 ↦ ((𝐴𝑥)𝑂(𝐵𝑥))) “ (V ∖ {𝑍})) ⊆ 𝐿)
2513, 24eqsstrd 3060 1 (𝜑 → ((𝐴𝑓 𝑂𝐵) “ (V ∖ {𝑍})) ⊆ 𝐿)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  wcel 1438  Vcvv 2619  cdif 2996  wss 2999  {csn 3446  cmpt 3899  ccnv 4437  cima 4441   Fn wfn 5010  wf 5011  cfv 5015  (class class class)co 5652  𝑓 cof 5854
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-setind 4353
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-of 5856
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator