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Theorem suppssov1 5761
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 2619 . . . . . . . 8 (𝐴𝑉𝐴 ∈ V)
31, 2syl 14 . . . . . . 7 ((𝜑𝑥𝐷) → 𝐴 ∈ V)
43adantr 270 . . . . . 6 (((𝜑𝑥𝐷) ∧ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})) → 𝐴 ∈ V)
5 eldifsni 3537 . . . . . . . 8 ((𝐴𝑂𝐵) ∈ (V ∖ {𝑍}) → (𝐴𝑂𝐵) ≠ 𝑍)
6 oveq2 5572 . . . . . . . . . . . 12 (𝑣 = 𝐵 → (𝑌𝑂𝑣) = (𝑌𝑂𝐵))
76eqeq1d 2091 . . . . . . . . . . 11 (𝑣 = 𝐵 → ((𝑌𝑂𝑣) = 𝑍 ↔ (𝑌𝑂𝐵) = 𝑍))
8 suppssov1.o . . . . . . . . . . . . 13 ((𝜑𝑣𝑅) → (𝑌𝑂𝑣) = 𝑍)
98ralrimiva 2439 . . . . . . . . . . . 12 (𝜑 → ∀𝑣𝑅 (𝑌𝑂𝑣) = 𝑍)
109adantr 270 . . . . . . . . . . 11 ((𝜑𝑥𝐷) → ∀𝑣𝑅 (𝑌𝑂𝑣) = 𝑍)
11 suppssov1.b . . . . . . . . . . 11 ((𝜑𝑥𝐷) → 𝐵𝑅)
127, 10, 11rspcdva 2715 . . . . . . . . . 10 ((𝜑𝑥𝐷) → (𝑌𝑂𝐵) = 𝑍)
13 oveq1 5571 . . . . . . . . . . 11 (𝐴 = 𝑌 → (𝐴𝑂𝐵) = (𝑌𝑂𝐵))
1413eqeq1d 2091 . . . . . . . . . 10 (𝐴 = 𝑌 → ((𝐴𝑂𝐵) = 𝑍 ↔ (𝑌𝑂𝐵) = 𝑍))
1512, 14syl5ibrcom 155 . . . . . . . . 9 ((𝜑𝑥𝐷) → (𝐴 = 𝑌 → (𝐴𝑂𝐵) = 𝑍))
1615necon3d 2293 . . . . . . . 8 ((𝜑𝑥𝐷) → ((𝐴𝑂𝐵) ≠ 𝑍𝐴𝑌))
175, 16syl5 32 . . . . . . 7 ((𝜑𝑥𝐷) → ((𝐴𝑂𝐵) ∈ (V ∖ {𝑍}) → 𝐴𝑌))
1817imp 122 . . . . . 6 (((𝜑𝑥𝐷) ∧ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})) → 𝐴𝑌)
19 eldifsn 3535 . . . . . 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 3084 . . 3 (𝜑 → {𝑥𝐷 ∣ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})} ⊆ {𝑥𝐷𝐴 ∈ (V ∖ {𝑌})})
23 eqid 2083 . . . 4 (𝑥𝐷 ↦ (𝐴𝑂𝐵)) = (𝑥𝐷 ↦ (𝐴𝑂𝐵))
2423mptpreima 4864 . . 3 ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) “ (V ∖ {𝑍})) = {𝑥𝐷 ∣ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})}
25 eqid 2083 . . . 4 (𝑥𝐷𝐴) = (𝑥𝐷𝐴)
2625mptpreima 4864 . . 3 ((𝑥𝐷𝐴) “ (V ∖ {𝑌})) = {𝑥𝐷𝐴 ∈ (V ∖ {𝑌})}
2722, 24, 263sstr4g 3049 . 2 (𝜑 → ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) “ (V ∖ {𝑍})) ⊆ ((𝑥𝐷𝐴) “ (V ∖ {𝑌})))
28 suppssov1.s . 2 (𝜑 → ((𝑥𝐷𝐴) “ (V ∖ {𝑌})) ⊆ 𝐿)
2927, 28sstrd 3018 1 (𝜑 → ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) “ (V ∖ {𝑍})) ⊆ 𝐿)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wcel 1434  wne 2249  wral 2353  {crab 2357  Vcvv 2610  cdif 2979  wss 2982  {csn 3416  cmpt 3859  ccnv 4390  cima 4394  (class class class)co 5564
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2612  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-mpt 3861  df-xp 4397  df-rel 4398  df-cnv 4399  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fv 4960  df-ov 5567
This theorem is referenced by:  suppssof1  5780
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