![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ttukeylem4 | Structured version Visualization version GIF version |
Description: Lemma for ttukey 10455. (Contributed by Mario Carneiro, 15-May-2015.) |
Ref | Expression |
---|---|
ttukeylem.1 | β’ (π β πΉ:(cardβ(βͺ π΄ β π΅))β1-1-ontoβ(βͺ π΄ β π΅)) |
ttukeylem.2 | β’ (π β π΅ β π΄) |
ttukeylem.3 | β’ (π β βπ₯(π₯ β π΄ β (π« π₯ β© Fin) β π΄)) |
ttukeylem.4 | β’ πΊ = recs((π§ β V β¦ if(dom π§ = βͺ dom π§, if(dom π§ = β , π΅, βͺ ran π§), ((π§ββͺ dom π§) βͺ if(((π§ββͺ dom π§) βͺ {(πΉββͺ dom π§)}) β π΄, {(πΉββͺ dom π§)}, β ))))) |
Ref | Expression |
---|---|
ttukeylem4 | β’ (π β (πΊββ ) = π΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0elon 6372 | . . 3 β’ β β On | |
2 | ttukeylem.1 | . . . 4 β’ (π β πΉ:(cardβ(βͺ π΄ β π΅))β1-1-ontoβ(βͺ π΄ β π΅)) | |
3 | ttukeylem.2 | . . . 4 β’ (π β π΅ β π΄) | |
4 | ttukeylem.3 | . . . 4 β’ (π β βπ₯(π₯ β π΄ β (π« π₯ β© Fin) β π΄)) | |
5 | ttukeylem.4 | . . . 4 β’ πΊ = recs((π§ β V β¦ if(dom π§ = βͺ dom π§, if(dom π§ = β , π΅, βͺ ran π§), ((π§ββͺ dom π§) βͺ if(((π§ββͺ dom π§) βͺ {(πΉββͺ dom π§)}) β π΄, {(πΉββͺ dom π§)}, β ))))) | |
6 | 2, 3, 4, 5 | ttukeylem3 10448 | . . 3 β’ ((π β§ β β On) β (πΊββ ) = if(β = βͺ β , if(β = β , π΅, βͺ (πΊ β β )), ((πΊββͺ β ) βͺ if(((πΊββͺ β ) βͺ {(πΉββͺ β )}) β π΄, {(πΉββͺ β )}, β )))) |
7 | 1, 6 | mpan2 690 | . 2 β’ (π β (πΊββ ) = if(β = βͺ β , if(β = β , π΅, βͺ (πΊ β β )), ((πΊββͺ β ) βͺ if(((πΊββͺ β ) βͺ {(πΉββͺ β )}) β π΄, {(πΉββͺ β )}, β )))) |
8 | uni0 4897 | . . . . 5 β’ βͺ β = β | |
9 | 8 | eqcomi 2746 | . . . 4 β’ β = βͺ β |
10 | 9 | iftruei 4494 | . . 3 β’ if(β = βͺ β , if(β = β , π΅, βͺ (πΊ β β )), ((πΊββͺ β ) βͺ if(((πΊββͺ β ) βͺ {(πΉββͺ β )}) β π΄, {(πΉββͺ β )}, β ))) = if(β = β , π΅, βͺ (πΊ β β )) |
11 | eqid 2737 | . . . 4 β’ β = β | |
12 | 11 | iftruei 4494 | . . 3 β’ if(β = β , π΅, βͺ (πΊ β β )) = π΅ |
13 | 10, 12 | eqtri 2765 | . 2 β’ if(β = βͺ β , if(β = β , π΅, βͺ (πΊ β β )), ((πΊββͺ β ) βͺ if(((πΊββͺ β ) βͺ {(πΉββͺ β )}) β π΄, {(πΉββͺ β )}, β ))) = π΅ |
14 | 7, 13 | eqtrdi 2793 | 1 β’ (π β (πΊββ ) = π΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 βwal 1540 = wceq 1542 β wcel 2107 Vcvv 3446 β cdif 3908 βͺ cun 3909 β© cin 3910 β wss 3911 β c0 4283 ifcif 4487 π« cpw 4561 {csn 4587 βͺ cuni 4866 β¦ cmpt 5189 dom cdm 5634 ran crn 5635 β cima 5637 Oncon0 6318 β1-1-ontoβwf1o 6496 βcfv 6497 recscrecs 8317 Fincfn 8884 cardccrd 9872 |
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 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 |
This theorem is referenced by: ttukeylem7 10452 |
Copyright terms: Public domain | W3C validator |