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Theorem satfun 34005
Description: The satisfaction predicate as function over wff codes in the model 𝑀 and the binary relation 𝐸 on 𝑀. (Contributed by AV, 29-Oct-2023.)
Assertion
Ref Expression
satfun ((𝑀𝑉𝐸𝑊) → ((𝑀 Sat 𝐸)‘ω):(Fmla‘ω)⟶𝒫 (𝑀m ω))

Proof of Theorem satfun
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 satff 34004 . . . . . 6 ((𝑀𝑉𝐸𝑊𝑥 ∈ ω) → ((𝑀 Sat 𝐸)‘𝑥):(Fmla‘𝑥)⟶𝒫 (𝑀m ω))
213expa 1118 . . . . 5 (((𝑀𝑉𝐸𝑊) ∧ 𝑥 ∈ ω) → ((𝑀 Sat 𝐸)‘𝑥):(Fmla‘𝑥)⟶𝒫 (𝑀m ω))
3 entric 10493 . . . . . . . . 9 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥𝑦𝑥𝑦𝑦𝑥))
43adantl 482 . . . . . . . 8 (((𝑀𝑉𝐸𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑥𝑦𝑥𝑦𝑦𝑥))
5 nnsdomo 9178 . . . . . . . . . . 11 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥𝑦𝑥𝑦))
65adantl 482 . . . . . . . . . 10 (((𝑀𝑉𝐸𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑥𝑦𝑥𝑦))
7 pm3.22 460 . . . . . . . . . . . . . 14 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑦 ∈ ω ∧ 𝑥 ∈ ω))
87anim2i 617 . . . . . . . . . . . . 13 (((𝑀𝑉𝐸𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → ((𝑀𝑉𝐸𝑊) ∧ (𝑦 ∈ ω ∧ 𝑥 ∈ ω)))
9 pssss 4055 . . . . . . . . . . . . 13 (𝑥𝑦𝑥𝑦)
10 eqid 2736 . . . . . . . . . . . . . . 15 (𝑀 Sat 𝐸) = (𝑀 Sat 𝐸)
1110satfsschain 33958 . . . . . . . . . . . . . 14 (((𝑀𝑉𝐸𝑊) ∧ (𝑦 ∈ ω ∧ 𝑥 ∈ ω)) → (𝑥𝑦 → ((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦)))
1211imp 407 . . . . . . . . . . . . 13 ((((𝑀𝑉𝐸𝑊) ∧ (𝑦 ∈ ω ∧ 𝑥 ∈ ω)) ∧ 𝑥𝑦) → ((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦))
138, 9, 12syl2an 596 . . . . . . . . . . . 12 ((((𝑀𝑉𝐸𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) ∧ 𝑥𝑦) → ((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦))
1413orcd 871 . . . . . . . . . . 11 ((((𝑀𝑉𝐸𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) ∧ 𝑥𝑦) → (((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦) ∨ ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥)))
1514ex 413 . . . . . . . . . 10 (((𝑀𝑉𝐸𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑥𝑦 → (((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦) ∨ ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥))))
166, 15sylbid 239 . . . . . . . . 9 (((𝑀𝑉𝐸𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑥𝑦 → (((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦) ∨ ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥))))
17 nneneq 9153 . . . . . . . . . . 11 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥𝑦𝑥 = 𝑦))
1817adantl 482 . . . . . . . . . 10 (((𝑀𝑉𝐸𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑥𝑦𝑥 = 𝑦))
19 ssid 3966 . . . . . . . . . . . 12 ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑦)
20 fveq2 6842 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝑀 Sat 𝐸)‘𝑥) = ((𝑀 Sat 𝐸)‘𝑦))
2119, 20sseqtrrid 3997 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥))
2221olcd 872 . . . . . . . . . 10 (𝑥 = 𝑦 → (((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦) ∨ ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥)))
2318, 22syl6bi 252 . . . . . . . . 9 (((𝑀𝑉𝐸𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑥𝑦 → (((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦) ∨ ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥))))
24 nnsdomo 9178 . . . . . . . . . . . 12 ((𝑦 ∈ ω ∧ 𝑥 ∈ ω) → (𝑦𝑥𝑦𝑥))
2524ancoms 459 . . . . . . . . . . 11 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑦𝑥𝑦𝑥))
2625adantl 482 . . . . . . . . . 10 (((𝑀𝑉𝐸𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑦𝑥𝑦𝑥))
2710satfsschain 33958 . . . . . . . . . . . . 13 (((𝑀𝑉𝐸𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑦𝑥 → ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥)))
28 pssss 4055 . . . . . . . . . . . . 13 (𝑦𝑥𝑦𝑥)
2927, 28impel 506 . . . . . . . . . . . 12 ((((𝑀𝑉𝐸𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) ∧ 𝑦𝑥) → ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥))
3029olcd 872 . . . . . . . . . . 11 ((((𝑀𝑉𝐸𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) ∧ 𝑦𝑥) → (((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦) ∨ ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥)))
3130ex 413 . . . . . . . . . 10 (((𝑀𝑉𝐸𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑦𝑥 → (((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦) ∨ ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥))))
3226, 31sylbid 239 . . . . . . . . 9 (((𝑀𝑉𝐸𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑦𝑥 → (((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦) ∨ ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥))))
3316, 23, 323jaod 1428 . . . . . . . 8 (((𝑀𝑉𝐸𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → ((𝑥𝑦𝑥𝑦𝑦𝑥) → (((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦) ∨ ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥))))
344, 33mpd 15 . . . . . . 7 (((𝑀𝑉𝐸𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦) ∨ ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥)))
3534expr 457 . . . . . 6 (((𝑀𝑉𝐸𝑊) ∧ 𝑥 ∈ ω) → (𝑦 ∈ ω → (((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦) ∨ ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥))))
3635ralrimiv 3142 . . . . 5 (((𝑀𝑉𝐸𝑊) ∧ 𝑥 ∈ ω) → ∀𝑦 ∈ ω (((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦) ∨ ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥)))
372, 36jca 512 . . . 4 (((𝑀𝑉𝐸𝑊) ∧ 𝑥 ∈ ω) → (((𝑀 Sat 𝐸)‘𝑥):(Fmla‘𝑥)⟶𝒫 (𝑀m ω) ∧ ∀𝑦 ∈ ω (((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦) ∨ ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥))))
3837ralrimiva 3143 . . 3 ((𝑀𝑉𝐸𝑊) → ∀𝑥 ∈ ω (((𝑀 Sat 𝐸)‘𝑥):(Fmla‘𝑥)⟶𝒫 (𝑀m ω) ∧ ∀𝑦 ∈ ω (((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦) ∨ ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥))))
39 fvex 6855 . . . 4 ((𝑀 Sat 𝐸)‘𝑥) ∈ V
4020, 39fiun 7875 . . 3 (∀𝑥 ∈ ω (((𝑀 Sat 𝐸)‘𝑥):(Fmla‘𝑥)⟶𝒫 (𝑀m ω) ∧ ∀𝑦 ∈ ω (((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦) ∨ ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥))) → 𝑥 ∈ ω ((𝑀 Sat 𝐸)‘𝑥): 𝑥 ∈ ω (Fmla‘𝑥)⟶𝒫 (𝑀m ω))
4138, 40syl 17 . 2 ((𝑀𝑉𝐸𝑊) → 𝑥 ∈ ω ((𝑀 Sat 𝐸)‘𝑥): 𝑥 ∈ ω (Fmla‘𝑥)⟶𝒫 (𝑀m ω))
42 satom 33950 . . 3 ((𝑀𝑉𝐸𝑊) → ((𝑀 Sat 𝐸)‘ω) = 𝑥 ∈ ω ((𝑀 Sat 𝐸)‘𝑥))
43 fmla 33975 . . . 4 (Fmla‘ω) = 𝑥 ∈ ω (Fmla‘𝑥)
4443a1i 11 . . 3 ((𝑀𝑉𝐸𝑊) → (Fmla‘ω) = 𝑥 ∈ ω (Fmla‘𝑥))
4542, 44feq12d 6656 . 2 ((𝑀𝑉𝐸𝑊) → (((𝑀 Sat 𝐸)‘ω):(Fmla‘ω)⟶𝒫 (𝑀m ω) ↔ 𝑥 ∈ ω ((𝑀 Sat 𝐸)‘𝑥): 𝑥 ∈ ω (Fmla‘𝑥)⟶𝒫 (𝑀m ω)))
4641, 45mpbird 256 1 ((𝑀𝑉𝐸𝑊) → ((𝑀 Sat 𝐸)‘ω):(Fmla‘ω)⟶𝒫 (𝑀m ω))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 845  w3o 1086   = wceq 1541  wcel 2106  wral 3064  wss 3910  wpss 3911  𝒫 cpw 4560   ciun 4954   class class class wbr 5105  wf 6492  cfv 6496  (class class class)co 7357  ωcom 7802  m cmap 8765  cen 8880  csdm 8882   Sat csat 33930  Fmlacfmla 33931
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-ac2 10399
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-ac 10052  df-goel 33934  df-gona 33935  df-goal 33936  df-sat 33937  df-fmla 33939
This theorem is referenced by:  satfvel  34006  satefvfmla0  34012  satefvfmla1  34019
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