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Theorem satfun 34700
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 ∈ π‘Š) β†’ ((𝑀 Sat 𝐸)β€˜Ο‰):(Fmlaβ€˜Ο‰)βŸΆπ’« (𝑀 ↑m Ο‰))

Proof of Theorem satfun
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 satff 34699 . . . . . 6 ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š ∧ π‘₯ ∈ Ο‰) β†’ ((𝑀 Sat 𝐸)β€˜π‘₯):(Fmlaβ€˜π‘₯)βŸΆπ’« (𝑀 ↑m Ο‰))
213expa 1116 . . . . 5 (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š) ∧ π‘₯ ∈ Ο‰) β†’ ((𝑀 Sat 𝐸)β€˜π‘₯):(Fmlaβ€˜π‘₯)βŸΆπ’« (𝑀 ↑m Ο‰))
3 entric 10554 . . . . . . . . 9 ((π‘₯ ∈ Ο‰ ∧ 𝑦 ∈ Ο‰) β†’ (π‘₯ β‰Ί 𝑦 ∨ π‘₯ β‰ˆ 𝑦 ∨ 𝑦 β‰Ί π‘₯))
43adantl 480 . . . . . . . 8 (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š) ∧ (π‘₯ ∈ Ο‰ ∧ 𝑦 ∈ Ο‰)) β†’ (π‘₯ β‰Ί 𝑦 ∨ π‘₯ β‰ˆ 𝑦 ∨ 𝑦 β‰Ί π‘₯))
5 nnsdomo 9236 . . . . . . . . . . 11 ((π‘₯ ∈ Ο‰ ∧ 𝑦 ∈ Ο‰) β†’ (π‘₯ β‰Ί 𝑦 ↔ π‘₯ ⊊ 𝑦))
65adantl 480 . . . . . . . . . 10 (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š) ∧ (π‘₯ ∈ Ο‰ ∧ 𝑦 ∈ Ο‰)) β†’ (π‘₯ β‰Ί 𝑦 ↔ π‘₯ ⊊ 𝑦))
7 pm3.22 458 . . . . . . . . . . . . . 14 ((π‘₯ ∈ Ο‰ ∧ 𝑦 ∈ Ο‰) β†’ (𝑦 ∈ Ο‰ ∧ π‘₯ ∈ Ο‰))
87anim2i 615 . . . . . . . . . . . . 13 (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š) ∧ (π‘₯ ∈ Ο‰ ∧ 𝑦 ∈ Ο‰)) β†’ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š) ∧ (𝑦 ∈ Ο‰ ∧ π‘₯ ∈ Ο‰)))
9 pssss 4094 . . . . . . . . . . . . 13 (π‘₯ ⊊ 𝑦 β†’ π‘₯ βŠ† 𝑦)
10 eqid 2730 . . . . . . . . . . . . . . 15 (𝑀 Sat 𝐸) = (𝑀 Sat 𝐸)
1110satfsschain 34653 . . . . . . . . . . . . . 14 (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š) ∧ (𝑦 ∈ Ο‰ ∧ π‘₯ ∈ Ο‰)) β†’ (π‘₯ βŠ† 𝑦 β†’ ((𝑀 Sat 𝐸)β€˜π‘₯) βŠ† ((𝑀 Sat 𝐸)β€˜π‘¦)))
1211imp 405 . . . . . . . . . . . . 13 ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š) ∧ (𝑦 ∈ Ο‰ ∧ π‘₯ ∈ Ο‰)) ∧ π‘₯ βŠ† 𝑦) β†’ ((𝑀 Sat 𝐸)β€˜π‘₯) βŠ† ((𝑀 Sat 𝐸)β€˜π‘¦))
138, 9, 12syl2an 594 . . . . . . . . . . . 12 ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š) ∧ (π‘₯ ∈ Ο‰ ∧ 𝑦 ∈ Ο‰)) ∧ π‘₯ ⊊ 𝑦) β†’ ((𝑀 Sat 𝐸)β€˜π‘₯) βŠ† ((𝑀 Sat 𝐸)β€˜π‘¦))
1413orcd 869 . . . . . . . . . . 11 ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š) ∧ (π‘₯ ∈ Ο‰ ∧ 𝑦 ∈ Ο‰)) ∧ π‘₯ ⊊ 𝑦) β†’ (((𝑀 Sat 𝐸)β€˜π‘₯) βŠ† ((𝑀 Sat 𝐸)β€˜π‘¦) ∨ ((𝑀 Sat 𝐸)β€˜π‘¦) βŠ† ((𝑀 Sat 𝐸)β€˜π‘₯)))
1514ex 411 . . . . . . . . . 10 (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š) ∧ (π‘₯ ∈ Ο‰ ∧ 𝑦 ∈ Ο‰)) β†’ (π‘₯ ⊊ 𝑦 β†’ (((𝑀 Sat 𝐸)β€˜π‘₯) βŠ† ((𝑀 Sat 𝐸)β€˜π‘¦) ∨ ((𝑀 Sat 𝐸)β€˜π‘¦) βŠ† ((𝑀 Sat 𝐸)β€˜π‘₯))))
166, 15sylbid 239 . . . . . . . . 9 (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š) ∧ (π‘₯ ∈ Ο‰ ∧ 𝑦 ∈ Ο‰)) β†’ (π‘₯ β‰Ί 𝑦 β†’ (((𝑀 Sat 𝐸)β€˜π‘₯) βŠ† ((𝑀 Sat 𝐸)β€˜π‘¦) ∨ ((𝑀 Sat 𝐸)β€˜π‘¦) βŠ† ((𝑀 Sat 𝐸)β€˜π‘₯))))
17 nneneq 9211 . . . . . . . . . . 11 ((π‘₯ ∈ Ο‰ ∧ 𝑦 ∈ Ο‰) β†’ (π‘₯ β‰ˆ 𝑦 ↔ π‘₯ = 𝑦))
1817adantl 480 . . . . . . . . . 10 (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š) ∧ (π‘₯ ∈ Ο‰ ∧ 𝑦 ∈ Ο‰)) β†’ (π‘₯ β‰ˆ 𝑦 ↔ π‘₯ = 𝑦))
19 ssid 4003 . . . . . . . . . . . 12 ((𝑀 Sat 𝐸)β€˜π‘¦) βŠ† ((𝑀 Sat 𝐸)β€˜π‘¦)
20 fveq2 6890 . . . . . . . . . . . 12 (π‘₯ = 𝑦 β†’ ((𝑀 Sat 𝐸)β€˜π‘₯) = ((𝑀 Sat 𝐸)β€˜π‘¦))
2119, 20sseqtrrid 4034 . . . . . . . . . . 11 (π‘₯ = 𝑦 β†’ ((𝑀 Sat 𝐸)β€˜π‘¦) βŠ† ((𝑀 Sat 𝐸)β€˜π‘₯))
2221olcd 870 . . . . . . . . . 10 (π‘₯ = 𝑦 β†’ (((𝑀 Sat 𝐸)β€˜π‘₯) βŠ† ((𝑀 Sat 𝐸)β€˜π‘¦) ∨ ((𝑀 Sat 𝐸)β€˜π‘¦) βŠ† ((𝑀 Sat 𝐸)β€˜π‘₯)))
2318, 22syl6bi 252 . . . . . . . . 9 (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š) ∧ (π‘₯ ∈ Ο‰ ∧ 𝑦 ∈ Ο‰)) β†’ (π‘₯ β‰ˆ 𝑦 β†’ (((𝑀 Sat 𝐸)β€˜π‘₯) βŠ† ((𝑀 Sat 𝐸)β€˜π‘¦) ∨ ((𝑀 Sat 𝐸)β€˜π‘¦) βŠ† ((𝑀 Sat 𝐸)β€˜π‘₯))))
24 nnsdomo 9236 . . . . . . . . . . . 12 ((𝑦 ∈ Ο‰ ∧ π‘₯ ∈ Ο‰) β†’ (𝑦 β‰Ί π‘₯ ↔ 𝑦 ⊊ π‘₯))
2524ancoms 457 . . . . . . . . . . 11 ((π‘₯ ∈ Ο‰ ∧ 𝑦 ∈ Ο‰) β†’ (𝑦 β‰Ί π‘₯ ↔ 𝑦 ⊊ π‘₯))
2625adantl 480 . . . . . . . . . 10 (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š) ∧ (π‘₯ ∈ Ο‰ ∧ 𝑦 ∈ Ο‰)) β†’ (𝑦 β‰Ί π‘₯ ↔ 𝑦 ⊊ π‘₯))
2710satfsschain 34653 . . . . . . . . . . . . 13 (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š) ∧ (π‘₯ ∈ Ο‰ ∧ 𝑦 ∈ Ο‰)) β†’ (𝑦 βŠ† π‘₯ β†’ ((𝑀 Sat 𝐸)β€˜π‘¦) βŠ† ((𝑀 Sat 𝐸)β€˜π‘₯)))
28 pssss 4094 . . . . . . . . . . . . 13 (𝑦 ⊊ π‘₯ β†’ 𝑦 βŠ† π‘₯)
2927, 28impel 504 . . . . . . . . . . . 12 ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š) ∧ (π‘₯ ∈ Ο‰ ∧ 𝑦 ∈ Ο‰)) ∧ 𝑦 ⊊ π‘₯) β†’ ((𝑀 Sat 𝐸)β€˜π‘¦) βŠ† ((𝑀 Sat 𝐸)β€˜π‘₯))
3029olcd 870 . . . . . . . . . . 11 ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š) ∧ (π‘₯ ∈ Ο‰ ∧ 𝑦 ∈ Ο‰)) ∧ 𝑦 ⊊ π‘₯) β†’ (((𝑀 Sat 𝐸)β€˜π‘₯) βŠ† ((𝑀 Sat 𝐸)β€˜π‘¦) ∨ ((𝑀 Sat 𝐸)β€˜π‘¦) βŠ† ((𝑀 Sat 𝐸)β€˜π‘₯)))
3130ex 411 . . . . . . . . . 10 (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š) ∧ (π‘₯ ∈ Ο‰ ∧ 𝑦 ∈ Ο‰)) β†’ (𝑦 ⊊ π‘₯ β†’ (((𝑀 Sat 𝐸)β€˜π‘₯) βŠ† ((𝑀 Sat 𝐸)β€˜π‘¦) ∨ ((𝑀 Sat 𝐸)β€˜π‘¦) βŠ† ((𝑀 Sat 𝐸)β€˜π‘₯))))
3226, 31sylbid 239 . . . . . . . . 9 (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š) ∧ (π‘₯ ∈ Ο‰ ∧ 𝑦 ∈ Ο‰)) β†’ (𝑦 β‰Ί π‘₯ β†’ (((𝑀 Sat 𝐸)β€˜π‘₯) βŠ† ((𝑀 Sat 𝐸)β€˜π‘¦) ∨ ((𝑀 Sat 𝐸)β€˜π‘¦) βŠ† ((𝑀 Sat 𝐸)β€˜π‘₯))))
3316, 23, 323jaod 1426 . . . . . . . 8 (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š) ∧ (π‘₯ ∈ Ο‰ ∧ 𝑦 ∈ Ο‰)) β†’ ((π‘₯ β‰Ί 𝑦 ∨ π‘₯ β‰ˆ 𝑦 ∨ 𝑦 β‰Ί π‘₯) β†’ (((𝑀 Sat 𝐸)β€˜π‘₯) βŠ† ((𝑀 Sat 𝐸)β€˜π‘¦) ∨ ((𝑀 Sat 𝐸)β€˜π‘¦) βŠ† ((𝑀 Sat 𝐸)β€˜π‘₯))))
344, 33mpd 15 . . . . . . 7 (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š) ∧ (π‘₯ ∈ Ο‰ ∧ 𝑦 ∈ Ο‰)) β†’ (((𝑀 Sat 𝐸)β€˜π‘₯) βŠ† ((𝑀 Sat 𝐸)β€˜π‘¦) ∨ ((𝑀 Sat 𝐸)β€˜π‘¦) βŠ† ((𝑀 Sat 𝐸)β€˜π‘₯)))
3534expr 455 . . . . . 6 (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š) ∧ π‘₯ ∈ Ο‰) β†’ (𝑦 ∈ Ο‰ β†’ (((𝑀 Sat 𝐸)β€˜π‘₯) βŠ† ((𝑀 Sat 𝐸)β€˜π‘¦) ∨ ((𝑀 Sat 𝐸)β€˜π‘¦) βŠ† ((𝑀 Sat 𝐸)β€˜π‘₯))))
3635ralrimiv 3143 . . . . 5 (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š) ∧ π‘₯ ∈ Ο‰) β†’ βˆ€π‘¦ ∈ Ο‰ (((𝑀 Sat 𝐸)β€˜π‘₯) βŠ† ((𝑀 Sat 𝐸)β€˜π‘¦) ∨ ((𝑀 Sat 𝐸)β€˜π‘¦) βŠ† ((𝑀 Sat 𝐸)β€˜π‘₯)))
372, 36jca 510 . . . 4 (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š) ∧ π‘₯ ∈ Ο‰) β†’ (((𝑀 Sat 𝐸)β€˜π‘₯):(Fmlaβ€˜π‘₯)βŸΆπ’« (𝑀 ↑m Ο‰) ∧ βˆ€π‘¦ ∈ Ο‰ (((𝑀 Sat 𝐸)β€˜π‘₯) βŠ† ((𝑀 Sat 𝐸)β€˜π‘¦) ∨ ((𝑀 Sat 𝐸)β€˜π‘¦) βŠ† ((𝑀 Sat 𝐸)β€˜π‘₯))))
3837ralrimiva 3144 . . 3 ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š) β†’ βˆ€π‘₯ ∈ Ο‰ (((𝑀 Sat 𝐸)β€˜π‘₯):(Fmlaβ€˜π‘₯)βŸΆπ’« (𝑀 ↑m Ο‰) ∧ βˆ€π‘¦ ∈ Ο‰ (((𝑀 Sat 𝐸)β€˜π‘₯) βŠ† ((𝑀 Sat 𝐸)β€˜π‘¦) ∨ ((𝑀 Sat 𝐸)β€˜π‘¦) βŠ† ((𝑀 Sat 𝐸)β€˜π‘₯))))
39 fvex 6903 . . . 4 ((𝑀 Sat 𝐸)β€˜π‘₯) ∈ V
4020, 39fiun 7931 . . 3 (βˆ€π‘₯ ∈ Ο‰ (((𝑀 Sat 𝐸)β€˜π‘₯):(Fmlaβ€˜π‘₯)βŸΆπ’« (𝑀 ↑m Ο‰) ∧ βˆ€π‘¦ ∈ Ο‰ (((𝑀 Sat 𝐸)β€˜π‘₯) βŠ† ((𝑀 Sat 𝐸)β€˜π‘¦) ∨ ((𝑀 Sat 𝐸)β€˜π‘¦) βŠ† ((𝑀 Sat 𝐸)β€˜π‘₯))) β†’ βˆͺ π‘₯ ∈ Ο‰ ((𝑀 Sat 𝐸)β€˜π‘₯):βˆͺ π‘₯ ∈ Ο‰ (Fmlaβ€˜π‘₯)βŸΆπ’« (𝑀 ↑m Ο‰))
4138, 40syl 17 . 2 ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š) β†’ βˆͺ π‘₯ ∈ Ο‰ ((𝑀 Sat 𝐸)β€˜π‘₯):βˆͺ π‘₯ ∈ Ο‰ (Fmlaβ€˜π‘₯)βŸΆπ’« (𝑀 ↑m Ο‰))
42 satom 34645 . . 3 ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š) β†’ ((𝑀 Sat 𝐸)β€˜Ο‰) = βˆͺ π‘₯ ∈ Ο‰ ((𝑀 Sat 𝐸)β€˜π‘₯))
43 fmla 34670 . . . 4 (Fmlaβ€˜Ο‰) = βˆͺ π‘₯ ∈ Ο‰ (Fmlaβ€˜π‘₯)
4443a1i 11 . . 3 ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š) β†’ (Fmlaβ€˜Ο‰) = βˆͺ π‘₯ ∈ Ο‰ (Fmlaβ€˜π‘₯))
4542, 44feq12d 6704 . 2 ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š) β†’ (((𝑀 Sat 𝐸)β€˜Ο‰):(Fmlaβ€˜Ο‰)βŸΆπ’« (𝑀 ↑m Ο‰) ↔ βˆͺ π‘₯ ∈ Ο‰ ((𝑀 Sat 𝐸)β€˜π‘₯):βˆͺ π‘₯ ∈ Ο‰ (Fmlaβ€˜π‘₯)βŸΆπ’« (𝑀 ↑m Ο‰)))
4641, 45mpbird 256 1 ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š) β†’ ((𝑀 Sat 𝐸)β€˜Ο‰):(Fmlaβ€˜Ο‰)βŸΆπ’« (𝑀 ↑m Ο‰))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 843   ∨ w3o 1084   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059   βŠ† wss 3947   ⊊ wpss 3948  π’« cpw 4601  βˆͺ ciun 4996   class class class wbr 5147  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411  Ο‰com 7857   ↑m cmap 8822   β‰ˆ cen 8938   β‰Ί csdm 8940   Sat csat 34625  Fmlacfmla 34626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-inf2 9638  ax-ac2 10460
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-card 9936  df-ac 10113  df-goel 34629  df-gona 34630  df-goal 34631  df-sat 34632  df-fmla 34634
This theorem is referenced by:  satfvel  34701  satefvfmla0  34707  satefvfmla1  34714
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