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Theorem suppssov1 5709
 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 2566 . . . . . . . 8 (𝐴𝑉𝐴 ∈ V)
31, 2syl 14 . . . . . . 7 ((𝜑𝑥𝐷) → 𝐴 ∈ V)
43adantr 261 . . . . . 6 (((𝜑𝑥𝐷) ∧ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})) → 𝐴 ∈ V)
5 eldifsni 3496 . . . . . . . 8 ((𝐴𝑂𝐵) ∈ (V ∖ {𝑍}) → (𝐴𝑂𝐵) ≠ 𝑍)
6 suppssov1.b . . . . . . . . . . 11 ((𝜑𝑥𝐷) → 𝐵𝑅)
7 suppssov1.o . . . . . . . . . . . . 13 ((𝜑𝑣𝑅) → (𝑌𝑂𝑣) = 𝑍)
87ralrimiva 2392 . . . . . . . . . . . 12 (𝜑 → ∀𝑣𝑅 (𝑌𝑂𝑣) = 𝑍)
98adantr 261 . . . . . . . . . . 11 ((𝜑𝑥𝐷) → ∀𝑣𝑅 (𝑌𝑂𝑣) = 𝑍)
10 oveq2 5520 . . . . . . . . . . . . 13 (𝑣 = 𝐵 → (𝑌𝑂𝑣) = (𝑌𝑂𝐵))
1110eqeq1d 2048 . . . . . . . . . . . 12 (𝑣 = 𝐵 → ((𝑌𝑂𝑣) = 𝑍 ↔ (𝑌𝑂𝐵) = 𝑍))
1211rspcva 2654 . . . . . . . . . . 11 ((𝐵𝑅 ∧ ∀𝑣𝑅 (𝑌𝑂𝑣) = 𝑍) → (𝑌𝑂𝐵) = 𝑍)
136, 9, 12syl2anc 391 . . . . . . . . . 10 ((𝜑𝑥𝐷) → (𝑌𝑂𝐵) = 𝑍)
14 oveq1 5519 . . . . . . . . . . 11 (𝐴 = 𝑌 → (𝐴𝑂𝐵) = (𝑌𝑂𝐵))
1514eqeq1d 2048 . . . . . . . . . 10 (𝐴 = 𝑌 → ((𝐴𝑂𝐵) = 𝑍 ↔ (𝑌𝑂𝐵) = 𝑍))
1613, 15syl5ibrcom 146 . . . . . . . . 9 ((𝜑𝑥𝐷) → (𝐴 = 𝑌 → (𝐴𝑂𝐵) = 𝑍))
1716necon3d 2249 . . . . . . . 8 ((𝜑𝑥𝐷) → ((𝐴𝑂𝐵) ≠ 𝑍𝐴𝑌))
185, 17syl5 28 . . . . . . 7 ((𝜑𝑥𝐷) → ((𝐴𝑂𝐵) ∈ (V ∖ {𝑍}) → 𝐴𝑌))
1918imp 115 . . . . . 6 (((𝜑𝑥𝐷) ∧ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})) → 𝐴𝑌)
20 eldifsn 3495 . . . . . 6 (𝐴 ∈ (V ∖ {𝑌}) ↔ (𝐴 ∈ V ∧ 𝐴𝑌))
214, 19, 20sylanbrc 394 . . . . 5 (((𝜑𝑥𝐷) ∧ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})) → 𝐴 ∈ (V ∖ {𝑌}))
2221ex 108 . . . 4 ((𝜑𝑥𝐷) → ((𝐴𝑂𝐵) ∈ (V ∖ {𝑍}) → 𝐴 ∈ (V ∖ {𝑌})))
2322ss2rabdv 3021 . . 3 (𝜑 → {𝑥𝐷 ∣ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})} ⊆ {𝑥𝐷𝐴 ∈ (V ∖ {𝑌})})
24 eqid 2040 . . . 4 (𝑥𝐷 ↦ (𝐴𝑂𝐵)) = (𝑥𝐷 ↦ (𝐴𝑂𝐵))
2524mptpreima 4814 . . 3 ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) “ (V ∖ {𝑍})) = {𝑥𝐷 ∣ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})}
26 eqid 2040 . . . 4 (𝑥𝐷𝐴) = (𝑥𝐷𝐴)
2726mptpreima 4814 . . 3 ((𝑥𝐷𝐴) “ (V ∖ {𝑌})) = {𝑥𝐷𝐴 ∈ (V ∖ {𝑌})}
2823, 25, 273sstr4g 2986 . 2 (𝜑 → ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) “ (V ∖ {𝑍})) ⊆ ((𝑥𝐷𝐴) “ (V ∖ {𝑌})))
29 suppssov1.s . 2 (𝜑 → ((𝑥𝐷𝐴) “ (V ∖ {𝑌})) ⊆ 𝐿)
3028, 29sstrd 2955 1 (𝜑 → ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) “ (V ∖ {𝑍})) ⊆ 𝐿)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393   ≠ wne 2204  ∀wral 2306  {crab 2310  Vcvv 2557   ∖ cdif 2914   ⊆ wss 2917  {csn 3375   ↦ cmpt 3818  ◡ccnv 4344   “ cima 4348  (class class class)co 5512 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-xp 4351  df-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fv 4910  df-ov 5515 This theorem is referenced by:  suppssof1  5728
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