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

Theorem satefv 32789
 Description: The simplified satisfaction predicate as function over wff codes in the model 𝑀 at the code 𝑈. (Contributed by AV, 30-Oct-2023.)
Assertion
Ref Expression
satefv ((𝑀𝑉𝑈𝑊) → (𝑀 Sat 𝑈) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑈))

Proof of Theorem satefv
Dummy variables 𝑚 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-sate 32719 . . 3 Sat = (𝑚 ∈ V, 𝑢 ∈ V ↦ (((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω)‘𝑢))
21a1i 11 . 2 ((𝑀𝑉𝑈𝑊) → Sat = (𝑚 ∈ V, 𝑢 ∈ V ↦ (((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω)‘𝑢)))
3 id 22 . . . . . . 7 (𝑚 = 𝑀𝑚 = 𝑀)
43sqxpeqd 5552 . . . . . . . 8 (𝑚 = 𝑀 → (𝑚 × 𝑚) = (𝑀 × 𝑀))
54ineq2d 4139 . . . . . . 7 (𝑚 = 𝑀 → ( E ∩ (𝑚 × 𝑚)) = ( E ∩ (𝑀 × 𝑀)))
63, 5oveq12d 7154 . . . . . 6 (𝑚 = 𝑀 → (𝑚 Sat ( E ∩ (𝑚 × 𝑚))) = (𝑀 Sat ( E ∩ (𝑀 × 𝑀))))
76fveq1d 6648 . . . . 5 (𝑚 = 𝑀 → ((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω) = ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω))
87adantr 484 . . . 4 ((𝑚 = 𝑀𝑢 = 𝑈) → ((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω) = ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω))
9 simpr 488 . . . 4 ((𝑚 = 𝑀𝑢 = 𝑈) → 𝑢 = 𝑈)
108, 9fveq12d 6653 . . 3 ((𝑚 = 𝑀𝑢 = 𝑈) → (((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω)‘𝑢) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑈))
1110adantl 485 . 2 (((𝑀𝑉𝑈𝑊) ∧ (𝑚 = 𝑀𝑢 = 𝑈)) → (((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω)‘𝑢) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑈))
12 elex 3459 . . 3 (𝑀𝑉𝑀 ∈ V)
1312adantr 484 . 2 ((𝑀𝑉𝑈𝑊) → 𝑀 ∈ V)
14 elex 3459 . . 3 (𝑈𝑊𝑈 ∈ V)
1514adantl 485 . 2 ((𝑀𝑉𝑈𝑊) → 𝑈 ∈ V)
16 fvexd 6661 . 2 ((𝑀𝑉𝑈𝑊) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑈) ∈ V)
172, 11, 13, 15, 16ovmpod 7283 1 ((𝑀𝑉𝑈𝑊) → (𝑀 Sat 𝑈) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑈))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ∩ cin 3880   E cep 5430   × cxp 5518  ‘cfv 6325  (class class class)co 7136   ∈ cmpo 7138  ωcom 7563   Sat csat 32711   Sat∈ csate 32713 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-iota 6284  df-fun 6327  df-fv 6333  df-ov 7139  df-oprab 7140  df-mpo 7141  df-sate 32719 This theorem is referenced by:  sate0  32790  satef  32791  satefvfmla0  32793  satefvfmla1  32800
 Copyright terms: Public domain W3C validator