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Theorem fpwwe 8510
 Description: Given any function from the powerset of to , canth2 7251 gives that the function is not injective, but we can say rather more than that. There is a unique well-ordered subset which "agrees" with in the sense that each initial segment maps to its upper bound, and such that the entire set maps to an element of the set (so that it cannot be extended without losing the well-ordering). This theorem can be used to prove dfac8a 7900. Theorem 1.1 of [KanamoriPincus] p. 415. (Contributed by Mario Carneiro, 18-May-2015.)
Hypotheses
Ref Expression
fpwwe.1
fpwwe.2
fpwwe.3
fpwwe.4
Assertion
Ref Expression
fpwwe
Distinct variable groups:   ,,   ,,,   ,,,   ,,,   ,,,   ,,,   ,,,
Allowed substitution hint:   ()

Proof of Theorem fpwwe
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 df-ov 6075 . . . . . 6
2 fo1st 6357 . . . . . . . 8
3 fofn 5646 . . . . . . . 8
42, 3ax-mp 8 . . . . . . 7
5 opex 4419 . . . . . . 7
6 fvco2 5789 . . . . . . 7
74, 5, 6mp2an 654 . . . . . 6
81, 7eqtri 2455 . . . . 5
9 fpwwe.1 . . . . . . . . 9
109relopabi 4991 . . . . . . . 8
11 brrelex12 4906 . . . . . . . 8
1210, 11mpan 652 . . . . . . 7
13 op1stg 6350 . . . . . . 7
1412, 13syl 16 . . . . . 6
1514fveq2d 5723 . . . . 5
168, 15syl5eq 2479 . . . 4
1716eleq1d 2501 . . 3
1817pm5.32i 619 . 2
19 vex 2951 . . . . . . . . . 10
20 cnvexg 5396 . . . . . . . . . 10
21 imaexg 5208 . . . . . . . . . 10
2219, 20, 21mp2b 10 . . . . . . . . 9
23 vex 2951 . . . . . . . . . . . 12
2419inex1 4336 . . . . . . . . . . . 12
2523, 24algrflem 6446 . . . . . . . . . . 11
26 fveq2 5719 . . . . . . . . . . 11
2725, 26syl5eq 2479 . . . . . . . . . 10
2827eqeq1d 2443 . . . . . . . . 9
2922, 28sbcie 3187 . . . . . . . 8
3029ralbii 2721 . . . . . . 7
3130anbi2i 676 . . . . . 6
3231anbi2i 676 . . . . 5
3332opabbii 4264 . . . 4
349, 33eqtr4i 2458 . . 3
35 fpwwe.2 . . 3
36 vex 2951 . . . . 5
3736, 19algrflem 6446 . . . 4
38 simp1 957 . . . . . . 7
3936elpw 3797 . . . . . . 7
4038, 39sylibr 204 . . . . . 6
41 19.8a 1762 . . . . . . . 8
42413ad2ant3 980 . . . . . . 7
43 ween 7905 . . . . . . 7
4442, 43sylibr 204 . . . . . 6
45 elin 3522 . . . . . 6
4640, 44, 45sylanbrc 646 . . . . 5
47 fpwwe.3 . . . . 5
4846, 47sylan2 461 . . . 4
4937, 48syl5eqel 2519 . . 3
50 fpwwe.4 . . 3
5134, 35, 49, 50fpwwe2 8507 . 2
5218, 51syl5bbr 251 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   w3a 936  wex 1550   wceq 1652   wcel 1725  wral 2697  cvv 2948  wsbc 3153   cin 3311   wss 3312  cpw 3791  csn 3806  cop 3809  cuni 4007   class class class wbr 4204  copab 4257   wwe 4532   cxp 4867  ccnv 4868   cdm 4869  cima 4872   ccom 4873   wrel 4874   wfn 5440  wfo 5443  cfv 5445  (class class class)co 6072  c1st 6338  ccrd 7811 This theorem is referenced by:  canth4  8511  canthnumlem  8512  canthp1lem2  8517 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-isom 5454  df-ov 6075  df-1st 6340  df-riota 6540  df-recs 6624  df-en 7101  df-oi 7468  df-card 7815
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