Theorem satfun 35149

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 35148	. . . . . 6 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑥 ∈ ω) → ((𝑀 Sat 𝐸)‘𝑥):(Fmla‘𝑥)⟶𝒫 (𝑀 ↑m ω))
2	1	3expa 1115	. . . . 5 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑥 ∈ ω) → ((𝑀 Sat 𝐸)‘𝑥):(Fmla‘𝑥)⟶𝒫 (𝑀 ↑m ω))
3		entric 10582	. . . . . . . . 9 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 ≺ 𝑦 ∨ 𝑥 ≈ 𝑦 ∨ 𝑦 ≺ 𝑥))
4	3	adantl 480	. . . . . . . 8 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑥 ≺ 𝑦 ∨ 𝑥 ≈ 𝑦 ∨ 𝑦 ≺ 𝑥))
5		nnsdomo 9259	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 ≺ 𝑦 ↔ 𝑥 ⊊ 𝑦))
6	5	adantl 480	. . . . . . . . . 10 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑥 ≺ 𝑦 ↔ 𝑥 ⊊ 𝑦))
7		pm3.22 458	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑦 ∈ ω ∧ 𝑥 ∈ ω))
8	7	anim2i 615	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑦 ∈ ω ∧ 𝑥 ∈ ω)))
9		pssss 4091	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊊ 𝑦 → 𝑥 ⊆ 𝑦)
10		eqid 2725	. . . . . . . . . . . . . . 15 ⊢ (𝑀 Sat 𝐸) = (𝑀 Sat 𝐸)
11	10	satfsschain 35102	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑦 ∈ ω ∧ 𝑥 ∈ ω)) → (𝑥 ⊆ 𝑦 → ((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦)))
12	11	imp 405	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑦 ∈ ω ∧ 𝑥 ∈ ω)) ∧ 𝑥 ⊆ 𝑦) → ((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦))
13	8, 9, 12	syl2an 594	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) ∧ 𝑥 ⊊ 𝑦) → ((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦))
14	13	orcd 871	. . . . . . . . . . 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 411	. . . . . . . . . 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 239	. . . . . . . . 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 9234	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 ≈ 𝑦 ↔ 𝑥 = 𝑦))
18	17	adantl 480	. . . . . . . . . 10 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑥 ≈ 𝑦 ↔ 𝑥 = 𝑦))
19		ssid 3999	. . . . . . . . . . . 12 ⊢ ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑦)
20		fveq2 6896	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((𝑀 Sat 𝐸)‘𝑥) = ((𝑀 Sat 𝐸)‘𝑦))
21	19, 20	sseqtrrid 4030	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥))
22	21	olcd 872	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦) ∨ ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥)))
23	18, 22	biimtrdi 252	. . . . . . . . 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 9259	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ 𝑥 ∈ ω) → (𝑦 ≺ 𝑥 ↔ 𝑦 ⊊ 𝑥))
25	24	ancoms 457	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑦 ≺ 𝑥 ↔ 𝑦 ⊊ 𝑥))
26	25	adantl 480	. . . . . . . . . 10 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑦 ≺ 𝑥 ↔ 𝑦 ⊊ 𝑥))
27	10	satfsschain 35102	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑦 ⊆ 𝑥 → ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥)))
28		pssss 4091	. . . . . . . . . . . . 13 ⊢ (𝑦 ⊊ 𝑥 → 𝑦 ⊆ 𝑥)
29	27, 28	impel 504	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) ∧ 𝑦 ⊊ 𝑥) → ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥))
30	29	olcd 872	. . . . . . . . . . 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 411	. . . . . . . . . 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 239	. . . . . . . . 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 1425	. . . . . . . 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 455	. . . . . 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 3134	. . . . 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 510	. . . 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 3135	. . 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 6909	. . . 4 ⊢ ((𝑀 Sat 𝐸)‘𝑥) ∈ V
40	20, 39	fiun 7947	. . 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 35094	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ((𝑀 Sat 𝐸)‘ω) = ∪ 𝑥 ∈ ω ((𝑀 Sat 𝐸)‘𝑥))
43		fmla 35119	. . . 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 6711	. 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 256	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ((𝑀 Sat 𝐸)‘ω):(Fmla‘ω)⟶𝒫 (𝑀 ↑m ω))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 845   ∨ w3o 1083   = wceq 1533   ∈ wcel 2098  ∀wral 3050   ⊆ wss 3944   ⊊ wpss 3945  𝒫 cpw 4604  ∪ ciun 4997   class class class wbr 5149  ⟶wf 6545  ‘cfv 6549  (class class class)co 7419  ωcom 7871   ↑_m cmap 8845   ≈ cen 8961   ≺ csdm 8963   Sat csat 35074  Fmlacfmla 35075

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 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-inf2 9666  ax-ac2 10488

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-2o 8488  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-card 9964  df-ac 10141  df-goel 35078  df-gona 35079  df-goal 35080  df-sat 35081  df-fmla 35083

This theorem is referenced by:  satfvel  35150  satefvfmla0  35156  satefvfmla1  35163