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Mirrors > Home > MPE Home > Th. List > Mathboxes > cardpred | Structured version Visualization version GIF version |
Description: The cardinality function preserves predecessors. (Contributed by BTernaryTau, 18-Jul-2024.) |
Ref | Expression |
---|---|
cardpred | β’ ((π΄ β dom card β§ π΅ β dom card) β Pred( E , (card β π΄), (cardβπ΅)) = (card β Pred( βΊ , π΄, π΅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cardf2 9976 | . . 3 β’ card:{π₯ β£ βπ¦ β On π¦ β π₯}βΆOn | |
2 | ffun 6730 | . . . 4 β’ (card:{π₯ β£ βπ¦ β On π¦ β π₯}βΆOn β Fun card) | |
3 | 2 | funfnd 6589 | . . 3 β’ (card:{π₯ β£ βπ¦ β On π¦ β π₯}βΆOn β card Fn dom card) |
4 | 1, 3 | mp1i 13 | . 2 β’ ((π΄ β dom card β§ π΅ β dom card) β card Fn dom card) |
5 | fvex 6915 | . . . . . 6 β’ (cardβπ¦) β V | |
6 | 5 | epeli 5588 | . . . . 5 β’ ((cardβπ₯) E (cardβπ¦) β (cardβπ₯) β (cardβπ¦)) |
7 | cardsdom2 10021 | . . . . 5 β’ ((π₯ β dom card β§ π¦ β dom card) β ((cardβπ₯) β (cardβπ¦) β π₯ βΊ π¦)) | |
8 | 6, 7 | bitr2id 283 | . . . 4 β’ ((π₯ β dom card β§ π¦ β dom card) β (π₯ βΊ π¦ β (cardβπ₯) E (cardβπ¦))) |
9 | 8 | rgen2 3195 | . . 3 β’ βπ₯ β dom cardβπ¦ β dom card(π₯ βΊ π¦ β (cardβπ₯) E (cardβπ¦)) |
10 | 9 | a1i 11 | . 2 β’ ((π΄ β dom card β§ π΅ β dom card) β βπ₯ β dom cardβπ¦ β dom card(π₯ βΊ π¦ β (cardβπ₯) E (cardβπ¦))) |
11 | simpl 481 | . 2 β’ ((π΄ β dom card β§ π΅ β dom card) β π΄ β dom card) | |
12 | simpr 483 | . 2 β’ ((π΄ β dom card β§ π΅ β dom card) β π΅ β dom card) | |
13 | 4, 10, 11, 12 | fnrelpredd 34753 | 1 β’ ((π΄ β dom card β§ π΅ β dom card) β Pred( E , (card β π΄), (cardβπ΅)) = (card β Pred( βΊ , π΄, π΅))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 {cab 2705 βwral 3058 βwrex 3067 β wss 3949 class class class wbr 5152 E cep 5585 dom cdm 5682 β cima 5685 Predcpred 6309 Oncon0 6374 Fn wfn 6548 βΆwf 6549 βcfv 6553 β cen 8969 βΊ csdm 8971 cardccrd 9968 |
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 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-card 9972 |
This theorem is referenced by: nummin 34755 |
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