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Mirrors > Home > MPE Home > Th. List > Mathboxes > onsucf1o | Structured version Visualization version GIF version |
Description: The successor operation is a bijective function between the ordinals and the class of succesor ordinals. Lemma 1.17 of [Schloeder] p. 2. (Contributed by RP, 18-Jan-2025.) |
Ref | Expression |
---|---|
onsucf1o.f | β’ πΉ = (π₯ β On β¦ suc π₯) |
Ref | Expression |
---|---|
onsucf1o | β’ πΉ:Onβ1-1-ontoβ{π β On β£ βπ β On π = suc π} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onsucf1o.f | . . . 4 β’ πΉ = (π₯ β On β¦ suc π₯) | |
2 | 1 | fin1a2lem2 10391 | . . 3 β’ πΉ:Onβ1-1βOn |
3 | f1fn 6778 | . . 3 β’ (πΉ:Onβ1-1βOn β πΉ Fn On) | |
4 | 2, 3 | ax-mp 5 | . 2 β’ πΉ Fn On |
5 | 1 | onsucrn 42476 | . 2 β’ ran πΉ = {π β On β£ βπ β On π = suc π} |
6 | 1 | fin1a2lem1 10390 | . . . . . 6 β’ (π β On β (πΉβπ) = suc π) |
7 | 1 | fin1a2lem1 10390 | . . . . . 6 β’ (π β On β (πΉβπ) = suc π) |
8 | 6, 7 | eqeqan12d 2738 | . . . . 5 β’ ((π β On β§ π β On) β ((πΉβπ) = (πΉβπ) β suc π = suc π)) |
9 | suc11 6461 | . . . . 5 β’ ((π β On β§ π β On) β (suc π = suc π β π = π)) | |
10 | 8, 9 | bitrd 279 | . . . 4 β’ ((π β On β§ π β On) β ((πΉβπ) = (πΉβπ) β π = π)) |
11 | 10 | biimpd 228 | . . 3 β’ ((π β On β§ π β On) β ((πΉβπ) = (πΉβπ) β π = π)) |
12 | 11 | rgen2 3189 | . 2 β’ βπ β On βπ β On ((πΉβπ) = (πΉβπ) β π = π) |
13 | dff1o6 7265 | . 2 β’ (πΉ:Onβ1-1-ontoβ{π β On β£ βπ β On π = suc π} β (πΉ Fn On β§ ran πΉ = {π β On β£ βπ β On π = suc π} β§ βπ β On βπ β On ((πΉβπ) = (πΉβπ) β π = π))) | |
14 | 4, 5, 12, 13 | mpbir3an 1338 | 1 β’ πΉ:Onβ1-1-ontoβ{π β On β£ βπ β On π = suc π} |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 βwral 3053 βwrex 3062 {crab 3424 β¦ cmpt 5221 ran crn 5667 Oncon0 6354 suc csuc 6356 Fn wfn 6528 β1-1βwf1 6530 β1-1-ontoβwf1o 6532 βcfv 6533 |
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 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 ax-un 7718 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-ord 6357 df-on 6358 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 |
This theorem is referenced by: (None) |
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