ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  suppssov1 GIF version

Theorem suppssov1 5931
Description: Formula building theorem for support restrictions: operator with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.)
Hypotheses
Ref Expression
suppssov1.s (𝜑 → ((𝑥𝐷𝐴) “ (V ∖ {𝑌})) ⊆ 𝐿)
suppssov1.o ((𝜑𝑣𝑅) → (𝑌𝑂𝑣) = 𝑍)
suppssov1.a ((𝜑𝑥𝐷) → 𝐴𝑉)
suppssov1.b ((𝜑𝑥𝐷) → 𝐵𝑅)
Assertion
Ref Expression
suppssov1 (𝜑 → ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) “ (V ∖ {𝑍})) ⊆ 𝐿)
Distinct variable groups:   𝜑,𝑣   𝜑,𝑥   𝑣,𝐵   𝑣,𝑂   𝑣,𝑅   𝑣,𝑌   𝑥,𝑌   𝑣,𝑍   𝑥,𝑍
Allowed substitution hints:   𝐴(𝑥,𝑣)   𝐵(𝑥)   𝐷(𝑥,𝑣)   𝑅(𝑥)   𝐿(𝑥,𝑣)   𝑂(𝑥)   𝑉(𝑥,𝑣)

Proof of Theorem suppssov1
StepHypRef Expression
1 suppssov1.a . . . . . . . 8 ((𝜑𝑥𝐷) → 𝐴𝑉)
2 elex 2666 . . . . . . . 8 (𝐴𝑉𝐴 ∈ V)
31, 2syl 14 . . . . . . 7 ((𝜑𝑥𝐷) → 𝐴 ∈ V)
43adantr 272 . . . . . 6 (((𝜑𝑥𝐷) ∧ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})) → 𝐴 ∈ V)
5 eldifsni 3616 . . . . . . . 8 ((𝐴𝑂𝐵) ∈ (V ∖ {𝑍}) → (𝐴𝑂𝐵) ≠ 𝑍)
6 oveq2 5734 . . . . . . . . . . . 12 (𝑣 = 𝐵 → (𝑌𝑂𝑣) = (𝑌𝑂𝐵))
76eqeq1d 2121 . . . . . . . . . . 11 (𝑣 = 𝐵 → ((𝑌𝑂𝑣) = 𝑍 ↔ (𝑌𝑂𝐵) = 𝑍))
8 suppssov1.o . . . . . . . . . . . . 13 ((𝜑𝑣𝑅) → (𝑌𝑂𝑣) = 𝑍)
98ralrimiva 2477 . . . . . . . . . . . 12 (𝜑 → ∀𝑣𝑅 (𝑌𝑂𝑣) = 𝑍)
109adantr 272 . . . . . . . . . . 11 ((𝜑𝑥𝐷) → ∀𝑣𝑅 (𝑌𝑂𝑣) = 𝑍)
11 suppssov1.b . . . . . . . . . . 11 ((𝜑𝑥𝐷) → 𝐵𝑅)
127, 10, 11rspcdva 2763 . . . . . . . . . 10 ((𝜑𝑥𝐷) → (𝑌𝑂𝐵) = 𝑍)
13 oveq1 5733 . . . . . . . . . . 11 (𝐴 = 𝑌 → (𝐴𝑂𝐵) = (𝑌𝑂𝐵))
1413eqeq1d 2121 . . . . . . . . . 10 (𝐴 = 𝑌 → ((𝐴𝑂𝐵) = 𝑍 ↔ (𝑌𝑂𝐵) = 𝑍))
1512, 14syl5ibrcom 156 . . . . . . . . 9 ((𝜑𝑥𝐷) → (𝐴 = 𝑌 → (𝐴𝑂𝐵) = 𝑍))
1615necon3d 2324 . . . . . . . 8 ((𝜑𝑥𝐷) → ((𝐴𝑂𝐵) ≠ 𝑍𝐴𝑌))
175, 16syl5 32 . . . . . . 7 ((𝜑𝑥𝐷) → ((𝐴𝑂𝐵) ∈ (V ∖ {𝑍}) → 𝐴𝑌))
1817imp 123 . . . . . 6 (((𝜑𝑥𝐷) ∧ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})) → 𝐴𝑌)
19 eldifsn 3614 . . . . . 6 (𝐴 ∈ (V ∖ {𝑌}) ↔ (𝐴 ∈ V ∧ 𝐴𝑌))
204, 18, 19sylanbrc 411 . . . . 5 (((𝜑𝑥𝐷) ∧ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})) → 𝐴 ∈ (V ∖ {𝑌}))
2120ex 114 . . . 4 ((𝜑𝑥𝐷) → ((𝐴𝑂𝐵) ∈ (V ∖ {𝑍}) → 𝐴 ∈ (V ∖ {𝑌})))
2221ss2rabdv 3142 . . 3 (𝜑 → {𝑥𝐷 ∣ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})} ⊆ {𝑥𝐷𝐴 ∈ (V ∖ {𝑌})})
23 eqid 2113 . . . 4 (𝑥𝐷 ↦ (𝐴𝑂𝐵)) = (𝑥𝐷 ↦ (𝐴𝑂𝐵))
2423mptpreima 4988 . . 3 ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) “ (V ∖ {𝑍})) = {𝑥𝐷 ∣ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})}
25 eqid 2113 . . . 4 (𝑥𝐷𝐴) = (𝑥𝐷𝐴)
2625mptpreima 4988 . . 3 ((𝑥𝐷𝐴) “ (V ∖ {𝑌})) = {𝑥𝐷𝐴 ∈ (V ∖ {𝑌})}
2722, 24, 263sstr4g 3104 . 2 (𝜑 → ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) “ (V ∖ {𝑍})) ⊆ ((𝑥𝐷𝐴) “ (V ∖ {𝑌})))
28 suppssov1.s . 2 (𝜑 → ((𝑥𝐷𝐴) “ (V ∖ {𝑌})) ⊆ 𝐿)
2927, 28sstrd 3071 1 (𝜑 → ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) “ (V ∖ {𝑍})) ⊆ 𝐿)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1312  wcel 1461  wne 2280  wral 2388  {crab 2392  Vcvv 2655  cdif 3032  wss 3035  {csn 3491  cmpt 3947  ccnv 4496  cima 4500  (class class class)co 5726
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-ral 2393  df-rex 2394  df-rab 2397  df-v 2657  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-mpt 3949  df-xp 4503  df-rel 4504  df-cnv 4505  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fv 5087  df-ov 5729
This theorem is referenced by:  suppssof1  5951
  Copyright terms: Public domain W3C validator