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Mirrors > Home > MPE Home > Th. List > ttukeylem4 | Structured version Visualization version GIF version |
Description: Lemma for ttukey 10541. (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 6418 | . . 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 10534 | . . 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 689 | . 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 4933 | . . . . 5 β’ βͺ β = β | |
9 | 8 | eqcomi 2734 | . . . 4 β’ β = βͺ β |
10 | 9 | iftruei 4531 | . . 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 2725 | . . . 4 β’ β = β | |
12 | 11 | iftruei 4531 | . . 3 β’ if(β = β , π΅, βͺ (πΊ β β )) = π΅ |
13 | 10, 12 | eqtri 2753 | . 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 2781 | 1 β’ (π β (πΊββ ) = π΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 βwal 1531 = wceq 1533 β wcel 2098 Vcvv 3463 β cdif 3936 βͺ cun 3937 β© cin 3938 β wss 3939 β c0 4318 ifcif 4524 π« cpw 4598 {csn 4624 βͺ cuni 4903 β¦ cmpt 5226 dom cdm 5672 ran crn 5673 β cima 5675 Oncon0 6364 β1-1-ontoβwf1o 6542 βcfv 6543 recscrecs 8389 Fincfn 8962 cardccrd 9958 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7419 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 |
This theorem is referenced by: ttukeylem7 10538 |
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