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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmla0 | Structured version Visualization version GIF version |
Description: The valid Godel formulas of height 0 is the set of all formulas of the form vi β vj ("Godel-set of membership") coded as β¨β , β¨π, πβ©β©. (Contributed by AV, 14-Sep-2023.) |
Ref | Expression |
---|---|
fmla0 | β’ (Fmlaββ ) = {π₯ β V β£ βπ β Ο βπ β Ο π₯ = (πβππ)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano1 7894 | . . 3 β’ β β Ο | |
2 | elelsuc 6442 | . . 3 β’ (β β Ο β β β suc Ο) | |
3 | fmlafv 34990 | . . 3 β’ (β β suc Ο β (Fmlaββ ) = dom ((β Sat β )ββ )) | |
4 | 1, 2, 3 | mp2b 10 | . 2 β’ (Fmlaββ ) = dom ((β Sat β )ββ ) |
5 | satf00 34984 | . . 3 β’ ((β Sat β )ββ ) = {β¨π₯, π¦β© β£ (π¦ = β β§ βπ β Ο βπ β Ο π₯ = (πβππ))} | |
6 | 5 | dmeqi 5907 | . 2 β’ dom ((β Sat β )ββ ) = dom {β¨π₯, π¦β© β£ (π¦ = β β§ βπ β Ο βπ β Ο π₯ = (πβππ))} |
7 | 0ex 5307 | . . . . . 6 β’ β β V | |
8 | 7 | isseti 3487 | . . . . 5 β’ βπ¦ π¦ = β |
9 | 19.41v 1946 | . . . . 5 β’ (βπ¦(π¦ = β β§ βπ β Ο βπ β Ο π₯ = (πβππ)) β (βπ¦ π¦ = β β§ βπ β Ο βπ β Ο π₯ = (πβππ))) | |
10 | 8, 9 | mpbiran 708 | . . . 4 β’ (βπ¦(π¦ = β β§ βπ β Ο βπ β Ο π₯ = (πβππ)) β βπ β Ο βπ β Ο π₯ = (πβππ)) |
11 | 10 | abbii 2798 | . . 3 β’ {π₯ β£ βπ¦(π¦ = β β§ βπ β Ο βπ β Ο π₯ = (πβππ))} = {π₯ β£ βπ β Ο βπ β Ο π₯ = (πβππ)} |
12 | dmopab 5918 | . . 3 β’ dom {β¨π₯, π¦β© β£ (π¦ = β β§ βπ β Ο βπ β Ο π₯ = (πβππ))} = {π₯ β£ βπ¦(π¦ = β β§ βπ β Ο βπ β Ο π₯ = (πβππ))} | |
13 | rabab 3500 | . . 3 β’ {π₯ β V β£ βπ β Ο βπ β Ο π₯ = (πβππ)} = {π₯ β£ βπ β Ο βπ β Ο π₯ = (πβππ)} | |
14 | 11, 12, 13 | 3eqtr4i 2766 | . 2 β’ dom {β¨π₯, π¦β© β£ (π¦ = β β§ βπ β Ο βπ β Ο π₯ = (πβππ))} = {π₯ β V β£ βπ β Ο βπ β Ο π₯ = (πβππ)} |
15 | 4, 6, 14 | 3eqtri 2760 | 1 β’ (Fmlaββ ) = {π₯ β V β£ βπ β Ο βπ β Ο π₯ = (πβππ)} |
Colors of variables: wff setvar class |
Syntax hints: β§ wa 395 = wceq 1534 βwex 1774 β wcel 2099 {cab 2705 βwrex 3067 {crab 3429 Vcvv 3471 β c0 4323 {copab 5210 dom cdm 5678 suc csuc 6371 βcfv 6548 (class class class)co 7420 Οcom 7870 βπcgoe 34943 Sat csat 34946 Fmlacfmla 34947 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9665 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 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 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-map 8847 df-goel 34950 df-sat 34953 df-fmla 34955 |
This theorem is referenced by: fmla0xp 34993 fmlafvel 34995 fmla1 34997 fmlaomn0 35000 gonan0 35002 goaln0 35003 gonar 35005 goalr 35007 fmla0disjsuc 35008 satfv0fvfmla0 35023 sategoelfvb 35029 |
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