![]() |
Mathbox for Mario Carneiro |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > satfdmfmla | Structured version Visualization version GIF version |
Description: The domain of the satisfaction predicate as function over wff codes in any model π and any binary relation πΈ on π for a natural number π is the set of valid Godel formulas of height π. (Contributed by AV, 13-Oct-2023.) |
Ref | Expression |
---|---|
satfdmfmla | β’ ((π β π β§ πΈ β π β§ π β Ο) β dom ((π Sat πΈ)βπ) = (Fmlaβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5306 | . . . . . . 7 β’ β β V | |
2 | 1, 1 | pm3.2i 469 | . . . . . 6 β’ (β β V β§ β β V) |
3 | 2 | jctr 523 | . . . . 5 β’ ((π β π β§ πΈ β π) β ((π β π β§ πΈ β π) β§ (β β V β§ β β V))) |
4 | 3 | 3adant3 1130 | . . . 4 β’ ((π β π β§ πΈ β π β§ π β Ο) β ((π β π β§ πΈ β π) β§ (β β V β§ β β V))) |
5 | satfdm 34658 | . . . 4 β’ (((π β π β§ πΈ β π) β§ (β β V β§ β β V)) β βπ β Ο dom ((π Sat πΈ)βπ) = dom ((β Sat β )βπ)) | |
6 | 4, 5 | syl 17 | . . 3 β’ ((π β π β§ πΈ β π β§ π β Ο) β βπ β Ο dom ((π Sat πΈ)βπ) = dom ((β Sat β )βπ)) |
7 | fveq2 6890 | . . . . . . 7 β’ (π = π β ((π Sat πΈ)βπ) = ((π Sat πΈ)βπ)) | |
8 | 7 | dmeqd 5904 | . . . . . 6 β’ (π = π β dom ((π Sat πΈ)βπ) = dom ((π Sat πΈ)βπ)) |
9 | fveq2 6890 | . . . . . . 7 β’ (π = π β ((β Sat β )βπ) = ((β Sat β )βπ)) | |
10 | 9 | dmeqd 5904 | . . . . . 6 β’ (π = π β dom ((β Sat β )βπ) = dom ((β Sat β )βπ)) |
11 | 8, 10 | eqeq12d 2746 | . . . . 5 β’ (π = π β (dom ((π Sat πΈ)βπ) = dom ((β Sat β )βπ) β dom ((π Sat πΈ)βπ) = dom ((β Sat β )βπ))) |
12 | 11 | rspcv 3607 | . . . 4 β’ (π β Ο β (βπ β Ο dom ((π Sat πΈ)βπ) = dom ((β Sat β )βπ) β dom ((π Sat πΈ)βπ) = dom ((β Sat β )βπ))) |
13 | 12 | 3ad2ant3 1133 | . . 3 β’ ((π β π β§ πΈ β π β§ π β Ο) β (βπ β Ο dom ((π Sat πΈ)βπ) = dom ((β Sat β )βπ) β dom ((π Sat πΈ)βπ) = dom ((β Sat β )βπ))) |
14 | 6, 13 | mpd 15 | . 2 β’ ((π β π β§ πΈ β π β§ π β Ο) β dom ((π Sat πΈ)βπ) = dom ((β Sat β )βπ)) |
15 | elelsuc 6436 | . . . 4 β’ (π β Ο β π β suc Ο) | |
16 | 15 | 3ad2ant3 1133 | . . 3 β’ ((π β π β§ πΈ β π β§ π β Ο) β π β suc Ο) |
17 | fmlafv 34669 | . . 3 β’ (π β suc Ο β (Fmlaβπ) = dom ((β Sat β )βπ)) | |
18 | 16, 17 | syl 17 | . 2 β’ ((π β π β§ πΈ β π β§ π β Ο) β (Fmlaβπ) = dom ((β Sat β )βπ)) |
19 | 14, 18 | eqtr4d 2773 | 1 β’ ((π β π β§ πΈ β π β§ π β Ο) β dom ((π Sat πΈ)βπ) = (Fmlaβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β§ w3a 1085 = wceq 1539 β wcel 2104 βwral 3059 Vcvv 3472 β c0 4321 dom cdm 5675 suc csuc 6365 βcfv 6542 (class class class)co 7411 Οcom 7857 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 |
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-ral 3060 df-rex 3069 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-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-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-goel 34629 df-goal 34631 df-sat 34632 df-fmla 34634 |
This theorem is referenced by: satffunlem1lem2 34692 satffunlem2lem2 34695 satff 34699 satefvfmla0 34707 satefvfmla1 34714 |
Copyright terms: Public domain | W3C validator |