Theorem satfun 35639

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	⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ((𝑀 Sat 𝐸)‘ω):(Fmla‘ω)⟶𝒫 (𝑀 ↑m ω))

Proof of Theorem satfun

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		satff 35638	. . . . . 6 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑥 ∈ ω) → ((𝑀 Sat 𝐸)‘𝑥):(Fmla‘𝑥)⟶𝒫 (𝑀 ↑m ω))
2	1	3expa 1124	. . . . 5 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑥 ∈ ω) → ((𝑀 Sat 𝐸)‘𝑥):(Fmla‘𝑥)⟶𝒫 (𝑀 ↑m ω))
3		entric 10470	. . . . . . . . 9 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 ≺ 𝑦 ∨ 𝑥 ≈ 𝑦 ∨ 𝑦 ≺ 𝑥))
4	3	adantl 482	. . . . . . . 8 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑥 ≺ 𝑦 ∨ 𝑥 ≈ 𝑦 ∨ 𝑦 ≺ 𝑥))
5		nnsdomo 9143	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 ≺ 𝑦 ↔ 𝑥 ⊊ 𝑦))
6	5	adantl 482	. . . . . . . . . 10 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑥 ≺ 𝑦 ↔ 𝑥 ⊊ 𝑦))
7		pm3.22 460	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑦 ∈ ω ∧ 𝑥 ∈ ω))
8	7	anim2i 623	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑦 ∈ ω ∧ 𝑥 ∈ ω)))
9		pssss 4029	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊊ 𝑦 → 𝑥 ⊆ 𝑦)
10		eqid 2739	. . . . . . . . . . . . . . 15 ⊢ (𝑀 Sat 𝐸) = (𝑀 Sat 𝐸)
11	10	satfsschain 35592	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑦 ∈ ω ∧ 𝑥 ∈ ω)) → (𝑥 ⊆ 𝑦 → ((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦)))
12	11	imp 407	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑦 ∈ ω ∧ 𝑥 ∈ ω)) ∧ 𝑥 ⊆ 𝑦) → ((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦))
13	8, 9, 12	syl2an 602	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) ∧ 𝑥 ⊊ 𝑦) → ((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦))
14	13	orcd 879	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) ∧ 𝑥 ⊊ 𝑦) → (((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦) ∨ ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥)))
15	14	ex 413	. . . . . . . . . 10 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑥 ⊊ 𝑦 → (((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦) ∨ ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥))))
16	6, 15	sylbid 241	. . . . . . . . 9 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑥 ≺ 𝑦 → (((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦) ∨ ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥))))
17		nneneq 9130	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 ≈ 𝑦 ↔ 𝑥 = 𝑦))
18	17	adantl 482	. . . . . . . . . 10 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑥 ≈ 𝑦 ↔ 𝑥 = 𝑦))
19		ssid 3937	. . . . . . . . . . . 12 ⊢ ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑦)
20		fveq2 6827	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((𝑀 Sat 𝐸)‘𝑥) = ((𝑀 Sat 𝐸)‘𝑦))
21	19, 20	sseqtrrid 3958	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥))
22	21	olcd 880	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦) ∨ ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥)))
23	18, 22	biimtrdi 254	. . . . . . . . 9 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑥 ≈ 𝑦 → (((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦) ∨ ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥))))
24		nnsdomo 9143	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ 𝑥 ∈ ω) → (𝑦 ≺ 𝑥 ↔ 𝑦 ⊊ 𝑥))
25	24	ancoms 459	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑦 ≺ 𝑥 ↔ 𝑦 ⊊ 𝑥))
26	25	adantl 482	. . . . . . . . . 10 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑦 ≺ 𝑥 ↔ 𝑦 ⊊ 𝑥))
27	10	satfsschain 35592	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑦 ⊆ 𝑥 → ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥)))
28		pssss 4029	. . . . . . . . . . . . 13 ⊢ (𝑦 ⊊ 𝑥 → 𝑦 ⊆ 𝑥)
29	27, 28	impel 510	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) ∧ 𝑦 ⊊ 𝑥) → ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥))
30	29	olcd 880	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) ∧ 𝑦 ⊊ 𝑥) → (((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦) ∨ ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥)))
31	30	ex 413	. . . . . . . . . 10 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑦 ⊊ 𝑥 → (((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦) ∨ ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥))))
32	26, 31	sylbid 241	. . . . . . . . 9 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑦 ≺ 𝑥 → (((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦) ∨ ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥))))
33	16, 23, 32	3jaod 1437	. . . . . . . 8 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → ((𝑥 ≺ 𝑦 ∨ 𝑥 ≈ 𝑦 ∨ 𝑦 ≺ 𝑥) → (((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦) ∨ ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥))))
34	4, 33	mpd 15	. . . . . . 7 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦) ∨ ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥)))
35	34	expr 457	. . . . . 6 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑥 ∈ ω) → (𝑦 ∈ ω → (((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦) ∨ ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥))))
36	35	ralrimiv 3130	. . . . 5 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑥 ∈ ω) → ∀𝑦 ∈ ω (((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦) ∨ ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥)))
37	2, 36	jca 516	. . . 4 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑥 ∈ ω) → (((𝑀 Sat 𝐸)‘𝑥):(Fmla‘𝑥)⟶𝒫 (𝑀 ↑m ω) ∧ ∀𝑦 ∈ ω (((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦) ∨ ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥))))
38	37	ralrimiva 3131	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ∀𝑥 ∈ ω (((𝑀 Sat 𝐸)‘𝑥):(Fmla‘𝑥)⟶𝒫 (𝑀 ↑m ω) ∧ ∀𝑦 ∈ ω (((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦) ∨ ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥))))
39		fvex 6840	. . . 4 ⊢ ((𝑀 Sat 𝐸)‘𝑥) ∈ V
40	20, 39	fiun 7885	. . 3 ⊢ (∀𝑥 ∈ ω (((𝑀 Sat 𝐸)‘𝑥):(Fmla‘𝑥)⟶𝒫 (𝑀 ↑m 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 (𝑀 ↑m ω))
41	38, 40	syl 17	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ∪ 𝑥 ∈ ω ((𝑀 Sat 𝐸)‘𝑥):∪ 𝑥 ∈ ω (Fmla‘𝑥)⟶𝒫 (𝑀 ↑m ω))
42		satom 35584	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ((𝑀 Sat 𝐸)‘ω) = ∪ 𝑥 ∈ ω ((𝑀 Sat 𝐸)‘𝑥))
43		fmla 35609	. . . 4 ⊢ (Fmla‘ω) = ∪ 𝑥 ∈ ω (Fmla‘𝑥)
44	43	a1i 11	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (Fmla‘ω) = ∪ 𝑥 ∈ ω (Fmla‘𝑥))
45	42, 44	feq12d 6643	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (((𝑀 Sat 𝐸)‘ω):(Fmla‘ω)⟶𝒫 (𝑀 ↑m ω) ↔ ∪ 𝑥 ∈ ω ((𝑀 Sat 𝐸)‘𝑥):∪ 𝑥 ∈ ω (Fmla‘𝑥)⟶𝒫 (𝑀 ↑m ω)))
46	41, 45	mpbird 258	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ((𝑀 Sat 𝐸)‘ω):(Fmla‘ω)⟶𝒫 (𝑀 ↑m ω))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853   ∨ w3o 1091   = wceq 1547   ∈ wcel 2119  ∀wral 3053   ⊆ wss 3883   ⊊ wpss 3884  𝒫 cpw 4529  ∪ ciun 4921   class class class wbr 5072  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356  ωcom 7806   ↑_m cmap 8763   ≈ cen 8880   ≺ csdm 8882   Sat csat 35564  Fmlacfmla 35565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-inf2 9553  ax-ac2 10376

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-se 5572  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-isom 6494  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-er 8633  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9854  df-ac 10029  df-goel 35568  df-gona 35569  df-goal 35570  df-sat 35571  df-fmla 35573

This theorem is referenced by:  satfvel  35640  satefvfmla0  35646  satefvfmla1  35653