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Mirrors > Home > MPE Home > Th. List > ttukeylem4 | Structured version Visualization version GIF version |
Description: Lemma for ttukey 10515. (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 6412 | . . 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 10508 | . . 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 688 | . 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 4932 | . . . . 5 β’ βͺ β = β | |
9 | 8 | eqcomi 2735 | . . . 4 β’ β = βͺ β |
10 | 9 | iftruei 4530 | . . 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 2726 | . . . 4 β’ β = β | |
12 | 11 | iftruei 4530 | . . 3 β’ if(β = β , π΅, βͺ (πΊ β β )) = π΅ |
13 | 10, 12 | eqtri 2754 | . 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 2782 | 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 3468 β cdif 3940 βͺ cun 3941 β© cin 3942 β wss 3943 β c0 4317 ifcif 4523 π« cpw 4597 {csn 4623 βͺ cuni 4902 β¦ cmpt 5224 dom cdm 5669 ran crn 5670 β cima 5672 Oncon0 6358 β1-1-ontoβwf1o 6536 βcfv 6537 recscrecs 8371 Fincfn 8941 cardccrd 9932 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 |
This theorem is referenced by: ttukeylem7 10512 |
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