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Theorem satfun 34402
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 34401 . . . . . 6 ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š ∧ π‘₯ ∈ Ο‰) β†’ ((𝑀 Sat 𝐸)β€˜π‘₯):(Fmlaβ€˜π‘₯)βŸΆπ’« (𝑀 ↑m Ο‰))
213expa 1119 . . . . 5 (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š) ∧ π‘₯ ∈ Ο‰) β†’ ((𝑀 Sat 𝐸)β€˜π‘₯):(Fmlaβ€˜π‘₯)βŸΆπ’« (𝑀 ↑m Ο‰))
3 entric 10552 . . . . . . . . 9 ((π‘₯ ∈ Ο‰ ∧ 𝑦 ∈ Ο‰) β†’ (π‘₯ β‰Ί 𝑦 ∨ π‘₯ β‰ˆ 𝑦 ∨ 𝑦 β‰Ί π‘₯))
43adantl 483 . . . . . . . 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 9234 . . . . . . . . . . 11 ((π‘₯ ∈ Ο‰ ∧ 𝑦 ∈ Ο‰) β†’ (π‘₯ β‰Ί 𝑦 ↔ π‘₯ ⊊ 𝑦))
65adantl 483 . . . . . . . . . 10 (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š) ∧ (π‘₯ ∈ Ο‰ ∧ 𝑦 ∈ Ο‰)) β†’ (π‘₯ β‰Ί 𝑦 ↔ π‘₯ ⊊ 𝑦))
7 pm3.22 461 . . . . . . . . . . . . . 14 ((π‘₯ ∈ Ο‰ ∧ 𝑦 ∈ Ο‰) β†’ (𝑦 ∈ Ο‰ ∧ π‘₯ ∈ Ο‰))
87anim2i 618 . . . . . . . . . . . . 13 (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š) ∧ (π‘₯ ∈ Ο‰ ∧ 𝑦 ∈ Ο‰)) β†’ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š) ∧ (𝑦 ∈ Ο‰ ∧ π‘₯ ∈ Ο‰)))
9 pssss 4096 . . . . . . . . . . . . 13 (π‘₯ ⊊ 𝑦 β†’ π‘₯ βŠ† 𝑦)
10 eqid 2733 . . . . . . . . . . . . . . 15 (𝑀 Sat 𝐸) = (𝑀 Sat 𝐸)
1110satfsschain 34355 . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . 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 597 . . . . . . . . . . . 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 872 . . . . . . . . . . 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 414 . . . . . . . . . 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 9209 . . . . . . . . . . 11 ((π‘₯ ∈ Ο‰ ∧ 𝑦 ∈ Ο‰) β†’ (π‘₯ β‰ˆ 𝑦 ↔ π‘₯ = 𝑦))
1817adantl 483 . . . . . . . . . 10 (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š) ∧ (π‘₯ ∈ Ο‰ ∧ 𝑦 ∈ Ο‰)) β†’ (π‘₯ β‰ˆ 𝑦 ↔ π‘₯ = 𝑦))
19 ssid 4005 . . . . . . . . . . . 12 ((𝑀 Sat 𝐸)β€˜π‘¦) βŠ† ((𝑀 Sat 𝐸)β€˜π‘¦)
20 fveq2 6892 . . . . . . . . . . . 12 (π‘₯ = 𝑦 β†’ ((𝑀 Sat 𝐸)β€˜π‘₯) = ((𝑀 Sat 𝐸)β€˜π‘¦))
2119, 20sseqtrrid 4036 . . . . . . . . . . 11 (π‘₯ = 𝑦 β†’ ((𝑀 Sat 𝐸)β€˜π‘¦) βŠ† ((𝑀 Sat 𝐸)β€˜π‘₯))
2221olcd 873 . . . . . . . . . 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 253 . . . . . . . . 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 9234 . . . . . . . . . . . 12 ((𝑦 ∈ Ο‰ ∧ π‘₯ ∈ Ο‰) β†’ (𝑦 β‰Ί π‘₯ ↔ 𝑦 ⊊ π‘₯))
2524ancoms 460 . . . . . . . . . . 11 ((π‘₯ ∈ Ο‰ ∧ 𝑦 ∈ Ο‰) β†’ (𝑦 β‰Ί π‘₯ ↔ 𝑦 ⊊ π‘₯))
2625adantl 483 . . . . . . . . . 10 (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š) ∧ (π‘₯ ∈ Ο‰ ∧ 𝑦 ∈ Ο‰)) β†’ (𝑦 β‰Ί π‘₯ ↔ 𝑦 ⊊ π‘₯))
2710satfsschain 34355 . . . . . . . . . . . . 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 4096 . . . . . . . . . . . . 13 (𝑦 ⊊ π‘₯ β†’ 𝑦 βŠ† π‘₯)
2927, 28impel 507 . . . . . . . . . . . 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 873 . . . . . . . . . . 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 414 . . . . . . . . . 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 1429 . . . . . . . 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 458 . . . . . 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 3146 . . . . 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 513 . . . 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 3147 . . 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 6905 . . . 4 ((𝑀 Sat 𝐸)β€˜π‘₯) ∈ V
4020, 39fiun 7929 . . 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 34347 . . 3 ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š) β†’ ((𝑀 Sat 𝐸)β€˜Ο‰) = βˆͺ π‘₯ ∈ Ο‰ ((𝑀 Sat 𝐸)β€˜π‘₯))
43 fmla 34372 . . . 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 6706 . 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 257 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 397   ∨ wo 846   ∨ w3o 1087   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062   βŠ† wss 3949   ⊊ wpss 3950  π’« cpw 4603  βˆͺ ciun 4998   class class class wbr 5149  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  Ο‰com 7855   ↑m cmap 8820   β‰ˆ cen 8936   β‰Ί csdm 8938   Sat csat 34327  Fmlacfmla 34328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-ac2 10458
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-2o 8467  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9934  df-ac 10111  df-goel 34331  df-gona 34332  df-goal 34333  df-sat 34334  df-fmla 34336
This theorem is referenced by:  satfvel  34403  satefvfmla0  34409  satefvfmla1  34416
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