Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  suppssfv Structured version   GIF version

Theorem suppssfv 5650
 Description: Formula building theorem for support restriction, on a function which preserves zero. (Contributed by Stefan O'Rear, 9-Mar-2015.)
Hypotheses
Ref Expression
suppssfv.a (φ → ((x 𝐷A) “ (V ∖ {𝑌})) ⊆ 𝐿)
suppssfv.f (φ → (𝐹𝑌) = 𝑍)
suppssfv.v ((φ x 𝐷) → A 𝑉)
Assertion
Ref Expression
suppssfv (φ → ((x 𝐷 ↦ (𝐹A)) “ (V ∖ {𝑍})) ⊆ 𝐿)
Distinct variable groups:   φ,x   x,𝑌   x,𝑍
Allowed substitution hints:   A(x)   𝐷(x)   𝐹(x)   𝐿(x)   𝑉(x)

Proof of Theorem suppssfv
StepHypRef Expression
1 eldifsni 3487 . . . . 5 ((𝐹A) (V ∖ {𝑍}) → (𝐹A) ≠ 𝑍)
2 suppssfv.v . . . . . . . . 9 ((φ x 𝐷) → A 𝑉)
3 elex 2560 . . . . . . . . 9 (A 𝑉A V)
42, 3syl 14 . . . . . . . 8 ((φ x 𝐷) → A V)
54adantr 261 . . . . . . 7 (((φ x 𝐷) (𝐹A) ≠ 𝑍) → A V)
6 suppssfv.f . . . . . . . . . . 11 (φ → (𝐹𝑌) = 𝑍)
7 fveq2 5121 . . . . . . . . . . . 12 (A = 𝑌 → (𝐹A) = (𝐹𝑌))
87eqeq1d 2045 . . . . . . . . . . 11 (A = 𝑌 → ((𝐹A) = 𝑍 ↔ (𝐹𝑌) = 𝑍))
96, 8syl5ibrcom 146 . . . . . . . . . 10 (φ → (A = 𝑌 → (𝐹A) = 𝑍))
109necon3d 2243 . . . . . . . . 9 (φ → ((𝐹A) ≠ 𝑍A𝑌))
1110adantr 261 . . . . . . . 8 ((φ x 𝐷) → ((𝐹A) ≠ 𝑍A𝑌))
1211imp 115 . . . . . . 7 (((φ x 𝐷) (𝐹A) ≠ 𝑍) → A𝑌)
13 eldifsn 3486 . . . . . . 7 (A (V ∖ {𝑌}) ↔ (A V A𝑌))
145, 12, 13sylanbrc 394 . . . . . 6 (((φ x 𝐷) (𝐹A) ≠ 𝑍) → A (V ∖ {𝑌}))
1514ex 108 . . . . 5 ((φ x 𝐷) → ((𝐹A) ≠ 𝑍A (V ∖ {𝑌})))
161, 15syl5 28 . . . 4 ((φ x 𝐷) → ((𝐹A) (V ∖ {𝑍}) → A (V ∖ {𝑌})))
1716ss2rabdv 3015 . . 3 (φ → {x 𝐷 ∣ (𝐹A) (V ∖ {𝑍})} ⊆ {x 𝐷A (V ∖ {𝑌})})
18 eqid 2037 . . . 4 (x 𝐷 ↦ (𝐹A)) = (x 𝐷 ↦ (𝐹A))
1918mptpreima 4757 . . 3 ((x 𝐷 ↦ (𝐹A)) “ (V ∖ {𝑍})) = {x 𝐷 ∣ (𝐹A) (V ∖ {𝑍})}
20 eqid 2037 . . . 4 (x 𝐷A) = (x 𝐷A)
2120mptpreima 4757 . . 3 ((x 𝐷A) “ (V ∖ {𝑌})) = {x 𝐷A (V ∖ {𝑌})}
2217, 19, 213sstr4g 2980 . 2 (φ → ((x 𝐷 ↦ (𝐹A)) “ (V ∖ {𝑍})) ⊆ ((x 𝐷A) “ (V ∖ {𝑌})))
23 suppssfv.a . 2 (φ → ((x 𝐷A) “ (V ∖ {𝑌})) ⊆ 𝐿)
2422, 23sstrd 2949 1 (φ → ((x 𝐷 ↦ (𝐹A)) “ (V ∖ {𝑍})) ⊆ 𝐿)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390   ≠ wne 2201  {crab 2304  Vcvv 2551   ∖ cdif 2908   ⊆ wss 2911  {csn 3367   ↦ cmpt 3809  ◡ccnv 4287   “ cima 4291  ‘cfv 4845 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 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator