Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isfmlasuc Structured version   Visualization version   GIF version

Theorem isfmlasuc 34907
Description: The characterization of a Godel formula of height at least 1. (Contributed by AV, 14-Oct-2023.)
Assertion
Ref Expression
isfmlasuc ((𝑁 ∈ Ο‰ ∧ 𝐹 ∈ 𝑉) β†’ (𝐹 ∈ (Fmlaβ€˜suc 𝑁) ↔ (𝐹 ∈ (Fmlaβ€˜π‘) ∨ βˆƒπ‘’ ∈ (Fmlaβ€˜π‘)(βˆƒπ‘£ ∈ (Fmlaβ€˜π‘)𝐹 = (π‘’βŠΌπ‘”π‘£) ∨ βˆƒπ‘– ∈ Ο‰ 𝐹 = βˆ€π‘”π‘–π‘’))))
Distinct variable groups:   𝑖,𝐹,𝑒,𝑣   𝑖,𝑁,𝑒,𝑣
Allowed substitution hints:   𝑉(𝑣,𝑒,𝑖)

Proof of Theorem isfmlasuc
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 fmlasuc 34905 . . . 4 (𝑁 ∈ Ο‰ β†’ (Fmlaβ€˜suc 𝑁) = ((Fmlaβ€˜π‘) βˆͺ {𝑓 ∣ βˆƒπ‘’ ∈ (Fmlaβ€˜π‘)(βˆƒπ‘£ ∈ (Fmlaβ€˜π‘)𝑓 = (π‘’βŠΌπ‘”π‘£) ∨ βˆƒπ‘– ∈ Ο‰ 𝑓 = βˆ€π‘”π‘–π‘’)}))
21adantr 480 . . 3 ((𝑁 ∈ Ο‰ ∧ 𝐹 ∈ 𝑉) β†’ (Fmlaβ€˜suc 𝑁) = ((Fmlaβ€˜π‘) βˆͺ {𝑓 ∣ βˆƒπ‘’ ∈ (Fmlaβ€˜π‘)(βˆƒπ‘£ ∈ (Fmlaβ€˜π‘)𝑓 = (π‘’βŠΌπ‘”π‘£) ∨ βˆƒπ‘– ∈ Ο‰ 𝑓 = βˆ€π‘”π‘–π‘’)}))
32eleq2d 2813 . 2 ((𝑁 ∈ Ο‰ ∧ 𝐹 ∈ 𝑉) β†’ (𝐹 ∈ (Fmlaβ€˜suc 𝑁) ↔ 𝐹 ∈ ((Fmlaβ€˜π‘) βˆͺ {𝑓 ∣ βˆƒπ‘’ ∈ (Fmlaβ€˜π‘)(βˆƒπ‘£ ∈ (Fmlaβ€˜π‘)𝑓 = (π‘’βŠΌπ‘”π‘£) ∨ βˆƒπ‘– ∈ Ο‰ 𝑓 = βˆ€π‘”π‘–π‘’)})))
4 elun 4143 . . 3 (𝐹 ∈ ((Fmlaβ€˜π‘) βˆͺ {𝑓 ∣ βˆƒπ‘’ ∈ (Fmlaβ€˜π‘)(βˆƒπ‘£ ∈ (Fmlaβ€˜π‘)𝑓 = (π‘’βŠΌπ‘”π‘£) ∨ βˆƒπ‘– ∈ Ο‰ 𝑓 = βˆ€π‘”π‘–π‘’)}) ↔ (𝐹 ∈ (Fmlaβ€˜π‘) ∨ 𝐹 ∈ {𝑓 ∣ βˆƒπ‘’ ∈ (Fmlaβ€˜π‘)(βˆƒπ‘£ ∈ (Fmlaβ€˜π‘)𝑓 = (π‘’βŠΌπ‘”π‘£) ∨ βˆƒπ‘– ∈ Ο‰ 𝑓 = βˆ€π‘”π‘–π‘’)}))
54a1i 11 . 2 ((𝑁 ∈ Ο‰ ∧ 𝐹 ∈ 𝑉) β†’ (𝐹 ∈ ((Fmlaβ€˜π‘) βˆͺ {𝑓 ∣ βˆƒπ‘’ ∈ (Fmlaβ€˜π‘)(βˆƒπ‘£ ∈ (Fmlaβ€˜π‘)𝑓 = (π‘’βŠΌπ‘”π‘£) ∨ βˆƒπ‘– ∈ Ο‰ 𝑓 = βˆ€π‘”π‘–π‘’)}) ↔ (𝐹 ∈ (Fmlaβ€˜π‘) ∨ 𝐹 ∈ {𝑓 ∣ βˆƒπ‘’ ∈ (Fmlaβ€˜π‘)(βˆƒπ‘£ ∈ (Fmlaβ€˜π‘)𝑓 = (π‘’βŠΌπ‘”π‘£) ∨ βˆƒπ‘– ∈ Ο‰ 𝑓 = βˆ€π‘”π‘–π‘’)})))
6 eqeq1 2730 . . . . . . . 8 (𝑓 = 𝐹 β†’ (𝑓 = (π‘’βŠΌπ‘”π‘£) ↔ 𝐹 = (π‘’βŠΌπ‘”π‘£)))
76rexbidv 3172 . . . . . . 7 (𝑓 = 𝐹 β†’ (βˆƒπ‘£ ∈ (Fmlaβ€˜π‘)𝑓 = (π‘’βŠΌπ‘”π‘£) ↔ βˆƒπ‘£ ∈ (Fmlaβ€˜π‘)𝐹 = (π‘’βŠΌπ‘”π‘£)))
8 eqeq1 2730 . . . . . . . 8 (𝑓 = 𝐹 β†’ (𝑓 = βˆ€π‘”π‘–π‘’ ↔ 𝐹 = βˆ€π‘”π‘–π‘’))
98rexbidv 3172 . . . . . . 7 (𝑓 = 𝐹 β†’ (βˆƒπ‘– ∈ Ο‰ 𝑓 = βˆ€π‘”π‘–π‘’ ↔ βˆƒπ‘– ∈ Ο‰ 𝐹 = βˆ€π‘”π‘–π‘’))
107, 9orbi12d 915 . . . . . 6 (𝑓 = 𝐹 β†’ ((βˆƒπ‘£ ∈ (Fmlaβ€˜π‘)𝑓 = (π‘’βŠΌπ‘”π‘£) ∨ βˆƒπ‘– ∈ Ο‰ 𝑓 = βˆ€π‘”π‘–π‘’) ↔ (βˆƒπ‘£ ∈ (Fmlaβ€˜π‘)𝐹 = (π‘’βŠΌπ‘”π‘£) ∨ βˆƒπ‘– ∈ Ο‰ 𝐹 = βˆ€π‘”π‘–π‘’)))
1110rexbidv 3172 . . . . 5 (𝑓 = 𝐹 β†’ (βˆƒπ‘’ ∈ (Fmlaβ€˜π‘)(βˆƒπ‘£ ∈ (Fmlaβ€˜π‘)𝑓 = (π‘’βŠΌπ‘”π‘£) ∨ βˆƒπ‘– ∈ Ο‰ 𝑓 = βˆ€π‘”π‘–π‘’) ↔ βˆƒπ‘’ ∈ (Fmlaβ€˜π‘)(βˆƒπ‘£ ∈ (Fmlaβ€˜π‘)𝐹 = (π‘’βŠΌπ‘”π‘£) ∨ βˆƒπ‘– ∈ Ο‰ 𝐹 = βˆ€π‘”π‘–π‘’)))
1211elabg 3661 . . . 4 (𝐹 ∈ 𝑉 β†’ (𝐹 ∈ {𝑓 ∣ βˆƒπ‘’ ∈ (Fmlaβ€˜π‘)(βˆƒπ‘£ ∈ (Fmlaβ€˜π‘)𝑓 = (π‘’βŠΌπ‘”π‘£) ∨ βˆƒπ‘– ∈ Ο‰ 𝑓 = βˆ€π‘”π‘–π‘’)} ↔ βˆƒπ‘’ ∈ (Fmlaβ€˜π‘)(βˆƒπ‘£ ∈ (Fmlaβ€˜π‘)𝐹 = (π‘’βŠΌπ‘”π‘£) ∨ βˆƒπ‘– ∈ Ο‰ 𝐹 = βˆ€π‘”π‘–π‘’)))
1312adantl 481 . . 3 ((𝑁 ∈ Ο‰ ∧ 𝐹 ∈ 𝑉) β†’ (𝐹 ∈ {𝑓 ∣ βˆƒπ‘’ ∈ (Fmlaβ€˜π‘)(βˆƒπ‘£ ∈ (Fmlaβ€˜π‘)𝑓 = (π‘’βŠΌπ‘”π‘£) ∨ βˆƒπ‘– ∈ Ο‰ 𝑓 = βˆ€π‘”π‘–π‘’)} ↔ βˆƒπ‘’ ∈ (Fmlaβ€˜π‘)(βˆƒπ‘£ ∈ (Fmlaβ€˜π‘)𝐹 = (π‘’βŠΌπ‘”π‘£) ∨ βˆƒπ‘– ∈ Ο‰ 𝐹 = βˆ€π‘”π‘–π‘’)))
1413orbi2d 912 . 2 ((𝑁 ∈ Ο‰ ∧ 𝐹 ∈ 𝑉) β†’ ((𝐹 ∈ (Fmlaβ€˜π‘) ∨ 𝐹 ∈ {𝑓 ∣ βˆƒπ‘’ ∈ (Fmlaβ€˜π‘)(βˆƒπ‘£ ∈ (Fmlaβ€˜π‘)𝑓 = (π‘’βŠΌπ‘”π‘£) ∨ βˆƒπ‘– ∈ Ο‰ 𝑓 = βˆ€π‘”π‘–π‘’)}) ↔ (𝐹 ∈ (Fmlaβ€˜π‘) ∨ βˆƒπ‘’ ∈ (Fmlaβ€˜π‘)(βˆƒπ‘£ ∈ (Fmlaβ€˜π‘)𝐹 = (π‘’βŠΌπ‘”π‘£) ∨ βˆƒπ‘– ∈ Ο‰ 𝐹 = βˆ€π‘”π‘–π‘’))))
153, 5, 143bitrd 305 1 ((𝑁 ∈ Ο‰ ∧ 𝐹 ∈ 𝑉) β†’ (𝐹 ∈ (Fmlaβ€˜suc 𝑁) ↔ (𝐹 ∈ (Fmlaβ€˜π‘) ∨ βˆƒπ‘’ ∈ (Fmlaβ€˜π‘)(βˆƒπ‘£ ∈ (Fmlaβ€˜π‘)𝐹 = (π‘’βŠΌπ‘”π‘£) ∨ βˆƒπ‘– ∈ Ο‰ 𝐹 = βˆ€π‘”π‘–π‘’))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 844   = wceq 1533   ∈ wcel 2098  {cab 2703  βˆƒwrex 3064   βˆͺ cun 3941  suc csuc 6360  β€˜cfv 6537  (class class class)co 7405  Ο‰com 7852  βŠΌπ‘”cgna 34853  βˆ€π‘”cgol 34854  Fmlacfmla 34856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-inf2 9638
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-map 8824  df-goel 34859  df-goal 34861  df-sat 34862  df-fmla 34864
This theorem is referenced by:  gonarlem  34913  gonar  34914  goalrlem  34915  goalr  34916  fmlasucdisj  34918
  Copyright terms: Public domain W3C validator