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Theorem suppssov1 5835
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 2630 . . . . . . . 8 (𝐴𝑉𝐴 ∈ V)
31, 2syl 14 . . . . . . 7 ((𝜑𝑥𝐷) → 𝐴 ∈ V)
43adantr 270 . . . . . 6 (((𝜑𝑥𝐷) ∧ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})) → 𝐴 ∈ V)
5 eldifsni 3564 . . . . . . . 8 ((𝐴𝑂𝐵) ∈ (V ∖ {𝑍}) → (𝐴𝑂𝐵) ≠ 𝑍)
6 oveq2 5642 . . . . . . . . . . . 12 (𝑣 = 𝐵 → (𝑌𝑂𝑣) = (𝑌𝑂𝐵))
76eqeq1d 2096 . . . . . . . . . . 11 (𝑣 = 𝐵 → ((𝑌𝑂𝑣) = 𝑍 ↔ (𝑌𝑂𝐵) = 𝑍))
8 suppssov1.o . . . . . . . . . . . . 13 ((𝜑𝑣𝑅) → (𝑌𝑂𝑣) = 𝑍)
98ralrimiva 2446 . . . . . . . . . . . 12 (𝜑 → ∀𝑣𝑅 (𝑌𝑂𝑣) = 𝑍)
109adantr 270 . . . . . . . . . . 11 ((𝜑𝑥𝐷) → ∀𝑣𝑅 (𝑌𝑂𝑣) = 𝑍)
11 suppssov1.b . . . . . . . . . . 11 ((𝜑𝑥𝐷) → 𝐵𝑅)
127, 10, 11rspcdva 2727 . . . . . . . . . 10 ((𝜑𝑥𝐷) → (𝑌𝑂𝐵) = 𝑍)
13 oveq1 5641 . . . . . . . . . . 11 (𝐴 = 𝑌 → (𝐴𝑂𝐵) = (𝑌𝑂𝐵))
1413eqeq1d 2096 . . . . . . . . . 10 (𝐴 = 𝑌 → ((𝐴𝑂𝐵) = 𝑍 ↔ (𝑌𝑂𝐵) = 𝑍))
1512, 14syl5ibrcom 155 . . . . . . . . 9 ((𝜑𝑥𝐷) → (𝐴 = 𝑌 → (𝐴𝑂𝐵) = 𝑍))
1615necon3d 2299 . . . . . . . 8 ((𝜑𝑥𝐷) → ((𝐴𝑂𝐵) ≠ 𝑍𝐴𝑌))
175, 16syl5 32 . . . . . . 7 ((𝜑𝑥𝐷) → ((𝐴𝑂𝐵) ∈ (V ∖ {𝑍}) → 𝐴𝑌))
1817imp 122 . . . . . 6 (((𝜑𝑥𝐷) ∧ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})) → 𝐴𝑌)
19 eldifsn 3562 . . . . . 6 (𝐴 ∈ (V ∖ {𝑌}) ↔ (𝐴 ∈ V ∧ 𝐴𝑌))
204, 18, 19sylanbrc 408 . . . . 5 (((𝜑𝑥𝐷) ∧ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})) → 𝐴 ∈ (V ∖ {𝑌}))
2120ex 113 . . . 4 ((𝜑𝑥𝐷) → ((𝐴𝑂𝐵) ∈ (V ∖ {𝑍}) → 𝐴 ∈ (V ∖ {𝑌})))
2221ss2rabdv 3100 . . 3 (𝜑 → {𝑥𝐷 ∣ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})} ⊆ {𝑥𝐷𝐴 ∈ (V ∖ {𝑌})})
23 eqid 2088 . . . 4 (𝑥𝐷 ↦ (𝐴𝑂𝐵)) = (𝑥𝐷 ↦ (𝐴𝑂𝐵))
2423mptpreima 4911 . . 3 ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) “ (V ∖ {𝑍})) = {𝑥𝐷 ∣ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})}
25 eqid 2088 . . . 4 (𝑥𝐷𝐴) = (𝑥𝐷𝐴)
2625mptpreima 4911 . . 3 ((𝑥𝐷𝐴) “ (V ∖ {𝑌})) = {𝑥𝐷𝐴 ∈ (V ∖ {𝑌})}
2722, 24, 263sstr4g 3065 . 2 (𝜑 → ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) “ (V ∖ {𝑍})) ⊆ ((𝑥𝐷𝐴) “ (V ∖ {𝑌})))
28 suppssov1.s . 2 (𝜑 → ((𝑥𝐷𝐴) “ (V ∖ {𝑌})) ⊆ 𝐿)
2927, 28sstrd 3033 1 (𝜑 → ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) “ (V ∖ {𝑍})) ⊆ 𝐿)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  wcel 1438  wne 2255  wral 2359  {crab 2363  Vcvv 2619  cdif 2994  wss 2997  {csn 3441  cmpt 3891  ccnv 4427  cima 4431  (class class class)co 5634
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-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  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-rab 2368  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-xp 4434  df-rel 4435  df-cnv 4436  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fv 5010  df-ov 5637
This theorem is referenced by:  suppssof1  5854
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