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Theorem suppssov1 6094
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 2760 . . . . . . . 8 (𝐴𝑉𝐴 ∈ V)
31, 2syl 14 . . . . . . 7 ((𝜑𝑥𝐷) → 𝐴 ∈ V)
43adantr 276 . . . . . 6 (((𝜑𝑥𝐷) ∧ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})) → 𝐴 ∈ V)
5 eldifsni 3733 . . . . . . . 8 ((𝐴𝑂𝐵) ∈ (V ∖ {𝑍}) → (𝐴𝑂𝐵) ≠ 𝑍)
6 oveq2 5896 . . . . . . . . . . . 12 (𝑣 = 𝐵 → (𝑌𝑂𝑣) = (𝑌𝑂𝐵))
76eqeq1d 2196 . . . . . . . . . . 11 (𝑣 = 𝐵 → ((𝑌𝑂𝑣) = 𝑍 ↔ (𝑌𝑂𝐵) = 𝑍))
8 suppssov1.o . . . . . . . . . . . . 13 ((𝜑𝑣𝑅) → (𝑌𝑂𝑣) = 𝑍)
98ralrimiva 2560 . . . . . . . . . . . 12 (𝜑 → ∀𝑣𝑅 (𝑌𝑂𝑣) = 𝑍)
109adantr 276 . . . . . . . . . . 11 ((𝜑𝑥𝐷) → ∀𝑣𝑅 (𝑌𝑂𝑣) = 𝑍)
11 suppssov1.b . . . . . . . . . . 11 ((𝜑𝑥𝐷) → 𝐵𝑅)
127, 10, 11rspcdva 2858 . . . . . . . . . 10 ((𝜑𝑥𝐷) → (𝑌𝑂𝐵) = 𝑍)
13 oveq1 5895 . . . . . . . . . . 11 (𝐴 = 𝑌 → (𝐴𝑂𝐵) = (𝑌𝑂𝐵))
1413eqeq1d 2196 . . . . . . . . . 10 (𝐴 = 𝑌 → ((𝐴𝑂𝐵) = 𝑍 ↔ (𝑌𝑂𝐵) = 𝑍))
1512, 14syl5ibrcom 157 . . . . . . . . 9 ((𝜑𝑥𝐷) → (𝐴 = 𝑌 → (𝐴𝑂𝐵) = 𝑍))
1615necon3d 2401 . . . . . . . 8 ((𝜑𝑥𝐷) → ((𝐴𝑂𝐵) ≠ 𝑍𝐴𝑌))
175, 16syl5 32 . . . . . . 7 ((𝜑𝑥𝐷) → ((𝐴𝑂𝐵) ∈ (V ∖ {𝑍}) → 𝐴𝑌))
1817imp 124 . . . . . 6 (((𝜑𝑥𝐷) ∧ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})) → 𝐴𝑌)
19 eldifsn 3731 . . . . . 6 (𝐴 ∈ (V ∖ {𝑌}) ↔ (𝐴 ∈ V ∧ 𝐴𝑌))
204, 18, 19sylanbrc 417 . . . . 5 (((𝜑𝑥𝐷) ∧ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})) → 𝐴 ∈ (V ∖ {𝑌}))
2120ex 115 . . . 4 ((𝜑𝑥𝐷) → ((𝐴𝑂𝐵) ∈ (V ∖ {𝑍}) → 𝐴 ∈ (V ∖ {𝑌})))
2221ss2rabdv 3248 . . 3 (𝜑 → {𝑥𝐷 ∣ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})} ⊆ {𝑥𝐷𝐴 ∈ (V ∖ {𝑌})})
23 eqid 2187 . . . 4 (𝑥𝐷 ↦ (𝐴𝑂𝐵)) = (𝑥𝐷 ↦ (𝐴𝑂𝐵))
2423mptpreima 5134 . . 3 ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) “ (V ∖ {𝑍})) = {𝑥𝐷 ∣ (𝐴𝑂𝐵) ∈ (V ∖ {𝑍})}
25 eqid 2187 . . . 4 (𝑥𝐷𝐴) = (𝑥𝐷𝐴)
2625mptpreima 5134 . . 3 ((𝑥𝐷𝐴) “ (V ∖ {𝑌})) = {𝑥𝐷𝐴 ∈ (V ∖ {𝑌})}
2722, 24, 263sstr4g 3210 . 2 (𝜑 → ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) “ (V ∖ {𝑍})) ⊆ ((𝑥𝐷𝐴) “ (V ∖ {𝑌})))
28 suppssov1.s . 2 (𝜑 → ((𝑥𝐷𝐴) “ (V ∖ {𝑌})) ⊆ 𝐿)
2927, 28sstrd 3177 1 (𝜑 → ((𝑥𝐷 ↦ (𝐴𝑂𝐵)) “ (V ∖ {𝑍})) ⊆ 𝐿)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1363  wcel 2158  wne 2357  wral 2465  {crab 2469  Vcvv 2749  cdif 3138  wss 3141  {csn 3604  cmpt 4076  ccnv 4637  cima 4641  (class class class)co 5888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-xp 4644  df-rel 4645  df-cnv 4646  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fv 5236  df-ov 5891
This theorem is referenced by:  suppssof1  6114
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