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Mirrors > Home > MPE Home > Th. List > ackbij1lem7 | Structured version Visualization version GIF version |
Description: Lemma for ackbij1 10182. (Contributed by Stefan O'Rear, 21-Nov-2014.) |
Ref | Expression |
---|---|
ackbij.f | β’ πΉ = (π₯ β (π« Ο β© Fin) β¦ (cardββͺ π¦ β π₯ ({π¦} Γ π« π¦))) |
Ref | Expression |
---|---|
ackbij1lem7 | β’ (π΄ β (π« Ο β© Fin) β (πΉβπ΄) = (cardββͺ π¦ β π΄ ({π¦} Γ π« π¦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iuneq1 4974 | . . 3 β’ (π₯ = π΄ β βͺ π¦ β π₯ ({π¦} Γ π« π¦) = βͺ π¦ β π΄ ({π¦} Γ π« π¦)) | |
2 | 1 | fveq2d 6850 | . 2 β’ (π₯ = π΄ β (cardββͺ π¦ β π₯ ({π¦} Γ π« π¦)) = (cardββͺ π¦ β π΄ ({π¦} Γ π« π¦))) |
3 | ackbij.f | . 2 β’ πΉ = (π₯ β (π« Ο β© Fin) β¦ (cardββͺ π¦ β π₯ ({π¦} Γ π« π¦))) | |
4 | fvex 6859 | . 2 β’ (cardββͺ π¦ β π΄ ({π¦} Γ π« π¦)) β V | |
5 | 2, 3, 4 | fvmpt 6952 | 1 β’ (π΄ β (π« Ο β© Fin) β (πΉβπ΄) = (cardββͺ π¦ β π΄ ({π¦} Γ π« π¦))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 β© cin 3913 π« cpw 4564 {csn 4590 βͺ ciun 4958 β¦ cmpt 5192 Γ cxp 5635 βcfv 6500 Οcom 7806 Fincfn 8889 cardccrd 9879 |
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 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-iota 6452 df-fun 6502 df-fv 6508 |
This theorem is referenced by: ackbij1lem8 10171 ackbij1lem9 10172 |
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