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Theorem suppssov1 5651
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 → ((x 𝐷A) “ (V ∖ {𝑌})) ⊆ 𝐿)
suppssov1.o ((φ v 𝑅) → (𝑌𝑂v) = 𝑍)
suppssov1.a ((φ x 𝐷) → A 𝑉)
suppssov1.b ((φ x 𝐷) → B 𝑅)
Assertion
Ref Expression
suppssov1 (φ → ((x 𝐷 ↦ (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 ((φ x 𝐷) → A 𝑉)
2 elex 2560 . . . . . . . 8 (A 𝑉A V)
31, 2syl 14 . . . . . . 7 ((φ x 𝐷) → A V)
43adantr 261 . . . . . 6 (((φ x 𝐷) (A𝑂B) (V ∖ {𝑍})) → A V)
5 eldifsni 3487 . . . . . . . 8 ((A𝑂B) (V ∖ {𝑍}) → (A𝑂B) ≠ 𝑍)
6 suppssov1.b . . . . . . . . . . 11 ((φ x 𝐷) → B 𝑅)
7 suppssov1.o . . . . . . . . . . . . 13 ((φ v 𝑅) → (𝑌𝑂v) = 𝑍)
87ralrimiva 2386 . . . . . . . . . . . 12 (φv 𝑅 (𝑌𝑂v) = 𝑍)
98adantr 261 . . . . . . . . . . 11 ((φ x 𝐷) → v 𝑅 (𝑌𝑂v) = 𝑍)
10 oveq2 5463 . . . . . . . . . . . . 13 (v = B → (𝑌𝑂v) = (𝑌𝑂B))
1110eqeq1d 2045 . . . . . . . . . . . 12 (v = B → ((𝑌𝑂v) = 𝑍 ↔ (𝑌𝑂B) = 𝑍))
1211rspcva 2648 . . . . . . . . . . 11 ((B 𝑅 v 𝑅 (𝑌𝑂v) = 𝑍) → (𝑌𝑂B) = 𝑍)
136, 9, 12syl2anc 391 . . . . . . . . . 10 ((φ x 𝐷) → (𝑌𝑂B) = 𝑍)
14 oveq1 5462 . . . . . . . . . . 11 (A = 𝑌 → (A𝑂B) = (𝑌𝑂B))
1514eqeq1d 2045 . . . . . . . . . 10 (A = 𝑌 → ((A𝑂B) = 𝑍 ↔ (𝑌𝑂B) = 𝑍))
1613, 15syl5ibrcom 146 . . . . . . . . 9 ((φ x 𝐷) → (A = 𝑌 → (A𝑂B) = 𝑍))
1716necon3d 2243 . . . . . . . 8 ((φ x 𝐷) → ((A𝑂B) ≠ 𝑍A𝑌))
185, 17syl5 28 . . . . . . 7 ((φ x 𝐷) → ((A𝑂B) (V ∖ {𝑍}) → A𝑌))
1918imp 115 . . . . . 6 (((φ x 𝐷) (A𝑂B) (V ∖ {𝑍})) → A𝑌)
20 eldifsn 3486 . . . . . 6 (A (V ∖ {𝑌}) ↔ (A V A𝑌))
214, 19, 20sylanbrc 394 . . . . 5 (((φ x 𝐷) (A𝑂B) (V ∖ {𝑍})) → A (V ∖ {𝑌}))
2221ex 108 . . . 4 ((φ x 𝐷) → ((A𝑂B) (V ∖ {𝑍}) → A (V ∖ {𝑌})))
2322ss2rabdv 3015 . . 3 (φ → {x 𝐷 ∣ (A𝑂B) (V ∖ {𝑍})} ⊆ {x 𝐷A (V ∖ {𝑌})})
24 eqid 2037 . . . 4 (x 𝐷 ↦ (A𝑂B)) = (x 𝐷 ↦ (A𝑂B))
2524mptpreima 4757 . . 3 ((x 𝐷 ↦ (A𝑂B)) “ (V ∖ {𝑍})) = {x 𝐷 ∣ (A𝑂B) (V ∖ {𝑍})}
26 eqid 2037 . . . 4 (x 𝐷A) = (x 𝐷A)
2726mptpreima 4757 . . 3 ((x 𝐷A) “ (V ∖ {𝑌})) = {x 𝐷A (V ∖ {𝑌})}
2823, 25, 273sstr4g 2980 . 2 (φ → ((x 𝐷 ↦ (A𝑂B)) “ (V ∖ {𝑍})) ⊆ ((x 𝐷A) “ (V ∖ {𝑌})))
29 suppssov1.s . 2 (φ → ((x 𝐷A) “ (V ∖ {𝑌})) ⊆ 𝐿)
3028, 29sstrd 2949 1 (φ → ((x 𝐷 ↦ (A𝑂B)) “ (V ∖ {𝑍})) ⊆ 𝐿)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  wne 2201  wral 2300  {crab 2304  Vcvv 2551  cdif 2908  wss 2911  {csn 3367  cmpt 3809  ccnv 4287  cima 4291  (class class class)co 5455
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fv 4853  df-ov 5458
This theorem is referenced by:  suppssof1  5670
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