Theorem satfun 35624

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 35623	. . . . . 6 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑥 ∈ ω) → ((𝑀 Sat 𝐸)‘𝑥):(Fmla‘𝑥)⟶𝒫 (𝑀 ↑m ω))
2	1	3expa 1119	. . . . 5 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑥 ∈ ω) → ((𝑀 Sat 𝐸)‘𝑥):(Fmla‘𝑥)⟶𝒫 (𝑀 ↑m ω))
3		entric 10479	. . . . . . . . 9 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 ≺ 𝑦 ∨ 𝑥 ≈ 𝑦 ∨ 𝑦 ≺ 𝑥))
4	3	adantl 481	. . . . . . . 8 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑥 ≺ 𝑦 ∨ 𝑥 ≈ 𝑦 ∨ 𝑦 ≺ 𝑥))
5		nnsdomo 9155	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 ≺ 𝑦 ↔ 𝑥 ⊊ 𝑦))
6	5	adantl 481	. . . . . . . . . 10 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑥 ≺ 𝑦 ↔ 𝑥 ⊊ 𝑦))
7		pm3.22 459	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑦 ∈ ω ∧ 𝑥 ∈ ω))
8	7	anim2i 618	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑦 ∈ ω ∧ 𝑥 ∈ ω)))
9		pssss 4052	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊊ 𝑦 → 𝑥 ⊆ 𝑦)
10		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (𝑀 Sat 𝐸) = (𝑀 Sat 𝐸)
11	10	satfsschain 35577	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑦 ∈ ω ∧ 𝑥 ∈ ω)) → (𝑥 ⊆ 𝑦 → ((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦)))
12	11	imp 406	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑦 ∈ ω ∧ 𝑥 ∈ ω)) ∧ 𝑥 ⊆ 𝑦) → ((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦))
13	8, 9, 12	syl2an 597	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) ∧ 𝑥 ⊊ 𝑦) → ((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦))
14	13	orcd 874	. . . . . . . . . . 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 412	. . . . . . . . . 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 240	. . . . . . . . 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 9142	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 ≈ 𝑦 ↔ 𝑥 = 𝑦))
18	17	adantl 481	. . . . . . . . . 10 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑥 ≈ 𝑦 ↔ 𝑥 = 𝑦))
19		ssid 3958	. . . . . . . . . . . 12 ⊢ ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑦)
20		fveq2 6842	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((𝑀 Sat 𝐸)‘𝑥) = ((𝑀 Sat 𝐸)‘𝑦))
21	19, 20	sseqtrrid 3979	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥))
22	21	olcd 875	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (((𝑀 Sat 𝐸)‘𝑥) ⊆ ((𝑀 Sat 𝐸)‘𝑦) ∨ ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥)))
23	18, 22	biimtrdi 253	. . . . . . . . 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 9155	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ 𝑥 ∈ ω) → (𝑦 ≺ 𝑥 ↔ 𝑦 ⊊ 𝑥))
25	24	ancoms 458	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑦 ≺ 𝑥 ↔ 𝑦 ⊊ 𝑥))
26	25	adantl 481	. . . . . . . . . 10 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑦 ≺ 𝑥 ↔ 𝑦 ⊊ 𝑥))
27	10	satfsschain 35577	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑦 ⊆ 𝑥 → ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥)))
28		pssss 4052	. . . . . . . . . . . . 13 ⊢ (𝑦 ⊊ 𝑥 → 𝑦 ⊆ 𝑥)
29	27, 28	impel 505	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) ∧ 𝑦 ⊊ 𝑥) → ((𝑀 Sat 𝐸)‘𝑦) ⊆ ((𝑀 Sat 𝐸)‘𝑥))
30	29	olcd 875	. . . . . . . . . . 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 412	. . . . . . . . . 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 240	. . . . . . . . 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 1432	. . . . . . . 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 456	. . . . . 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 3129	. . . . 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 511	. . . 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 3130	. . 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 6855	. . . 4 ⊢ ((𝑀 Sat 𝐸)‘𝑥) ∈ V
40	20, 39	fiun 7897	. . 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 35569	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ((𝑀 Sat 𝐸)‘ω) = ∪ 𝑥 ∈ ω ((𝑀 Sat 𝐸)‘𝑥))
43		fmla 35594	. . . 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 6658	. 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 257	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ((𝑀 Sat 𝐸)‘ω):(Fmla‘ω)⟶𝒫 (𝑀 ↑m ω))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∨ w3o 1086   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ⊆ wss 3903   ⊊ wpss 3904  𝒫 cpw 4556  ∪ ciun 4948   class class class wbr 5100  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368  ωcom 7818   ↑_m cmap 8775   ≈ cen 8892   ≺ csdm 8894   Sat csat 35549  Fmlacfmla 35550

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-inf2 9562  ax-ac2 10385

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-er 8645  df-map 8777  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-card 9863  df-ac 10038  df-goel 35553  df-gona 35554  df-goal 35555  df-sat 35556  df-fmla 35558

This theorem is referenced by:  satfvel  35625  satefvfmla0  35631  satefvfmla1  35638