Theorem satfun 36384

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 36383	. . . . . 6 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑥 ∈ ω) → ((𝑀 Sat 𝐸)‘𝑥):(Fmla‘𝑥)⟶𝒫 (𝑀 ↑m ω))
2	1	3expa 1136	. . . . 5 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑥 ∈ ω) → ((𝑀 Sat 𝐸)‘𝑥):(Fmla‘𝑥)⟶𝒫 (𝑀 ↑m ω))
3		entric 10680	. . . . . . . . 9 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 ≺ 𝑦 ∨ 𝑥 ≈ 𝑦 ∨ 𝑦 ≺ 𝑥))
4	3	adantl 487	. . . . . . . 8 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑥 ≺ 𝑦 ∨ 𝑥 ≈ 𝑦 ∨ 𝑦 ≺ 𝑥))
5		nnsdomo 9272	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 ≺ 𝑦 ↔ 𝑥 ⊊ 𝑦))
6	5	adantl 487	. . . . . . . . . 10 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑥 ≺ 𝑦 ↔ 𝑥 ⊊ 𝑦))
7		pm3.22 465	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑦 ∈ ω ∧ 𝑥 ∈ ω))
8	7	anim2i 629	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑦 ∈ ω ∧ 𝑥 ∈ ω)))
9		pssss 4028	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊊ 𝑦 → 𝑥 ⊆ 𝑦)
10		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ (𝑀 Sat 𝐸) = (𝑀 Sat 𝐸)
11	10	satfsschain 36337	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑦 ∈ ω ∧ 𝑥 ∈ ω)) → (𝑥 ⊆ 𝑦 → ((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦)))
12	11	imp 412	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑦 ∈ ω ∧ 𝑥 ∈ ω)) ∧ 𝑥 ⊆ 𝑦) → ((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦))
13	8, 9, 12	syl2an 608	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) ∧ 𝑥 ⊊ 𝑦) → ((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦))
14	13	orcd 887	. . . . . . . . . . 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 418	. . . . . . . . . 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 243	. . . . . . . . 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 9259	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 ≈ 𝑦 ↔ 𝑥 = 𝑦))
18	17	adantl 487	. . . . . . . . . 10 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑥 ≈ 𝑦 ↔ 𝑥 = 𝑦))
19		ssid 3935	. . . . . . . . . . . 12 ⊢ ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑦)
20		fveq2 6916	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((𝑀 Sat 𝐸)‘𝑥) = ((𝑀 Sat 𝐸)‘𝑦))
21	19, 20	sseqtrrid 3956	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥))
22	21	olcd 888	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦) ∨ ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥)))
23	18, 22	biimtrdi 256	. . . . . . . . 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 9272	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ 𝑥 ∈ ω) → (𝑦 ≺ 𝑥 ↔ 𝑦 ⊊ 𝑥))
25	24	ancoms 464	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑦 ≺ 𝑥 ↔ 𝑦 ⊊ 𝑥))
26	25	adantl 487	. . . . . . . . . 10 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑦 ≺ 𝑥 ↔ 𝑦 ⊊ 𝑥))
27	10	satfsschain 36337	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑦 ⊆ 𝑥 → ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥)))
28		pssss 4028	. . . . . . . . . . . . 13 ⊢ (𝑦 ⊊ 𝑥 → 𝑦 ⊆ 𝑥)
29	27, 28	impel 515	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) ∧ 𝑦 ⊊ 𝑥) → ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥))
30	29	olcd 888	. . . . . . . . . . 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 418	. . . . . . . . . 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 243	. . . . . . . . 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 1457	. . . . . . . 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 16	. . . . . . 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 462	. . . . . 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 3157	. . . . 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 521	. . . 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 3158	. . 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 6929	. . . 4 ⊢ ((𝑀 Sat 𝐸)‘𝑥) ∈ V
40	20, 39	fiun 7989	. . 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 18	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ∪ 𝑥 ∈ ω ((𝑀 Sat 𝐸)‘𝑥):∪ 𝑥 ∈ ω (Fmla‘𝑥)⟶𝒫 (𝑀 ↑m ω))
42		satom 36329	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ((𝑀 Sat 𝐸)‘ω) = ∪ 𝑥 ∈ ω ((𝑀 Sat 𝐸)‘𝑥))
43		fmla 36354	. . . 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 6727	. 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 260	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ((𝑀 Sat 𝐸)‘ω):(Fmla‘ω)⟶𝒫 (𝑀 ↑m ω))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∨ w3o 1102   = wceq 1570   ∈ wcel 2147  ∀wral 3080   ⊆ wss 3900   ⊊ wpss 4016  𝒫 cpw 4560  ∪ ciun 4956   class class class wbr 5108  ⟶wf 6562  ‘cfv 6566  (class class class)co 7451  ωcom 7911   ↑_m cmap 8885   ≈ cen 9008   ≺ csdm 9010   Sat csat 36309  Fmlacfmla 36310

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-inf2 9679  ax-ac2 10580

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-isom 6575  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-2o 8515  df-er 8755  df-map 8887  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-card 10057  df-ac 10234  df-goel 36313  df-gona 36314  df-goal 36315  df-sat 36316  df-fmla 36318

This theorem is used by:  satfvel  36385  satefvfmla0  36391  satefvfmla1  36398