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Theorem fpwwe 10641
Description: Given any function š¹ from the powerset of š“ to š“, canth2 9130 gives that the function is not injective, but we can say rather more than that. There is a unique well-ordered subset āŸØš‘‹, (š‘Šā€˜š‘‹)āŸ© which "agrees" with š¹ 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 10025. Theorem 1.1 of [KanamoriPincus] p. 415. (Contributed by Mario Carneiro, 18-May-2015.) (Revised by AV, 20-Jul-2024.)
Hypotheses
Ref Expression
fpwwe.1 š‘Š = {āŸØš‘„, š‘ŸāŸ© āˆ£ ((š‘„ āŠ† š“ āˆ§ š‘Ÿ āŠ† (š‘„ Ɨ š‘„)) āˆ§ (š‘Ÿ We š‘„ āˆ§ āˆ€š‘¦ āˆˆ š‘„ (š¹ā€˜(ā—”š‘Ÿ ā€œ {š‘¦})) = š‘¦))}
fpwwe.2 (šœ‘ ā†’ š“ āˆˆ š‘‰)
fpwwe.3 ((šœ‘ āˆ§ š‘„ āˆˆ (š’« š“ āˆ© dom card)) ā†’ (š¹ā€˜š‘„) āˆˆ š“)
fpwwe.4 š‘‹ = āˆŖ dom š‘Š
Assertion
Ref Expression
fpwwe (šœ‘ ā†’ ((š‘Œš‘Šš‘… āˆ§ (š¹ā€˜š‘Œ) āˆˆ š‘Œ) ā†” (š‘Œ = š‘‹ āˆ§ š‘… = (š‘Šā€˜š‘‹))))
Distinct variable groups:   š‘„,š‘Ÿ,š“   š‘¦,š‘Ÿ,š¹,š‘„   šœ‘,š‘Ÿ,š‘„,š‘¦   š‘…,š‘Ÿ,š‘„,š‘¦   š‘‹,š‘Ÿ,š‘„,š‘¦   š‘Œ,š‘Ÿ,š‘„,š‘¦   š‘Š,š‘Ÿ,š‘„,š‘¦
Allowed substitution hints:   š“(š‘¦)   š‘‰(š‘„,š‘¦,š‘Ÿ)

Proof of Theorem fpwwe
Dummy variable š‘¢ is distinct from all other variables.
StepHypRef Expression
1 df-ov 7412 . . . . . 6 (š‘Œ(š¹ āˆ˜ 1st )š‘…) = ((š¹ āˆ˜ 1st )ā€˜āŸØš‘Œ, š‘…āŸ©)
2 fo1st 7995 . . . . . . . 8 1st :Vā€“ontoā†’V
3 fofn 6808 . . . . . . . 8 (1st :Vā€“ontoā†’V ā†’ 1st Fn V)
42, 3ax-mp 5 . . . . . . 7 1st Fn V
5 opex 5465 . . . . . . 7 āŸØš‘Œ, š‘…āŸ© āˆˆ V
6 fvco2 6989 . . . . . . 7 ((1st Fn V āˆ§ āŸØš‘Œ, š‘…āŸ© āˆˆ V) ā†’ ((š¹ āˆ˜ 1st )ā€˜āŸØš‘Œ, š‘…āŸ©) = (š¹ā€˜(1st ā€˜āŸØš‘Œ, š‘…āŸ©)))
74, 5, 6mp2an 691 . . . . . 6 ((š¹ āˆ˜ 1st )ā€˜āŸØš‘Œ, š‘…āŸ©) = (š¹ā€˜(1st ā€˜āŸØš‘Œ, š‘…āŸ©))
81, 7eqtri 2761 . . . . 5 (š‘Œ(š¹ āˆ˜ 1st )š‘…) = (š¹ā€˜(1st ā€˜āŸØš‘Œ, š‘…āŸ©))
9 fpwwe.1 . . . . . . . 8 š‘Š = {āŸØš‘„, š‘ŸāŸ© āˆ£ ((š‘„ āŠ† š“ āˆ§ š‘Ÿ āŠ† (š‘„ Ɨ š‘„)) āˆ§ (š‘Ÿ We š‘„ āˆ§ āˆ€š‘¦ āˆˆ š‘„ (š¹ā€˜(ā—”š‘Ÿ ā€œ {š‘¦})) = š‘¦))}
109bropaex12 5768 . . . . . . 7 (š‘Œš‘Šš‘… ā†’ (š‘Œ āˆˆ V āˆ§ š‘… āˆˆ V))
11 op1stg 7987 . . . . . . 7 ((š‘Œ āˆˆ V āˆ§ š‘… āˆˆ V) ā†’ (1st ā€˜āŸØš‘Œ, š‘…āŸ©) = š‘Œ)
1210, 11syl 17 . . . . . 6 (š‘Œš‘Šš‘… ā†’ (1st ā€˜āŸØš‘Œ, š‘…āŸ©) = š‘Œ)
1312fveq2d 6896 . . . . 5 (š‘Œš‘Šš‘… ā†’ (š¹ā€˜(1st ā€˜āŸØš‘Œ, š‘…āŸ©)) = (š¹ā€˜š‘Œ))
148, 13eqtrid 2785 . . . 4 (š‘Œš‘Šš‘… ā†’ (š‘Œ(š¹ āˆ˜ 1st )š‘…) = (š¹ā€˜š‘Œ))
1514eleq1d 2819 . . 3 (š‘Œš‘Šš‘… ā†’ ((š‘Œ(š¹ āˆ˜ 1st )š‘…) āˆˆ š‘Œ ā†” (š¹ā€˜š‘Œ) āˆˆ š‘Œ))
1615pm5.32i 576 . 2 ((š‘Œš‘Šš‘… āˆ§ (š‘Œ(š¹ āˆ˜ 1st )š‘…) āˆˆ š‘Œ) ā†” (š‘Œš‘Šš‘… āˆ§ (š¹ā€˜š‘Œ) āˆˆ š‘Œ))
17 vex 3479 . . . . . . . . . . 11 š‘Ÿ āˆˆ V
1817cnvex 7916 . . . . . . . . . 10 ā—”š‘Ÿ āˆˆ V
1918imaex 7907 . . . . . . . . 9 (ā—”š‘Ÿ ā€œ {š‘¦}) āˆˆ V
20 vex 3479 . . . . . . . . . . . 12 š‘¢ āˆˆ V
2117inex1 5318 . . . . . . . . . . . 12 (š‘Ÿ āˆ© (š‘¢ Ɨ š‘¢)) āˆˆ V
2220, 21opco1i 8111 . . . . . . . . . . 11 (š‘¢(š¹ āˆ˜ 1st )(š‘Ÿ āˆ© (š‘¢ Ɨ š‘¢))) = (š¹ā€˜š‘¢)
23 fveq2 6892 . . . . . . . . . . 11 (š‘¢ = (ā—”š‘Ÿ ā€œ {š‘¦}) ā†’ (š¹ā€˜š‘¢) = (š¹ā€˜(ā—”š‘Ÿ ā€œ {š‘¦})))
2422, 23eqtrid 2785 . . . . . . . . . 10 (š‘¢ = (ā—”š‘Ÿ ā€œ {š‘¦}) ā†’ (š‘¢(š¹ āˆ˜ 1st )(š‘Ÿ āˆ© (š‘¢ Ɨ š‘¢))) = (š¹ā€˜(ā—”š‘Ÿ ā€œ {š‘¦})))
2524eqeq1d 2735 . . . . . . . . 9 (š‘¢ = (ā—”š‘Ÿ ā€œ {š‘¦}) ā†’ ((š‘¢(š¹ āˆ˜ 1st )(š‘Ÿ āˆ© (š‘¢ Ɨ š‘¢))) = š‘¦ ā†” (š¹ā€˜(ā—”š‘Ÿ ā€œ {š‘¦})) = š‘¦))
2619, 25sbcie 3821 . . . . . . . 8 ([(ā—”š‘Ÿ ā€œ {š‘¦}) / š‘¢](š‘¢(š¹ āˆ˜ 1st )(š‘Ÿ āˆ© (š‘¢ Ɨ š‘¢))) = š‘¦ ā†” (š¹ā€˜(ā—”š‘Ÿ ā€œ {š‘¦})) = š‘¦)
2726ralbii 3094 . . . . . . 7 (āˆ€š‘¦ āˆˆ š‘„ [(ā—”š‘Ÿ ā€œ {š‘¦}) / š‘¢](š‘¢(š¹ āˆ˜ 1st )(š‘Ÿ āˆ© (š‘¢ Ɨ š‘¢))) = š‘¦ ā†” āˆ€š‘¦ āˆˆ š‘„ (š¹ā€˜(ā—”š‘Ÿ ā€œ {š‘¦})) = š‘¦)
2827anbi2i 624 . . . . . 6 ((š‘Ÿ We š‘„ āˆ§ āˆ€š‘¦ āˆˆ š‘„ [(ā—”š‘Ÿ ā€œ {š‘¦}) / š‘¢](š‘¢(š¹ āˆ˜ 1st )(š‘Ÿ āˆ© (š‘¢ Ɨ š‘¢))) = š‘¦) ā†” (š‘Ÿ We š‘„ āˆ§ āˆ€š‘¦ āˆˆ š‘„ (š¹ā€˜(ā—”š‘Ÿ ā€œ {š‘¦})) = š‘¦))
2928anbi2i 624 . . . . 5 (((š‘„ āŠ† š“ āˆ§ š‘Ÿ āŠ† (š‘„ Ɨ š‘„)) āˆ§ (š‘Ÿ We š‘„ āˆ§ āˆ€š‘¦ āˆˆ š‘„ [(ā—”š‘Ÿ ā€œ {š‘¦}) / š‘¢](š‘¢(š¹ āˆ˜ 1st )(š‘Ÿ āˆ© (š‘¢ Ɨ š‘¢))) = š‘¦)) ā†” ((š‘„ āŠ† š“ āˆ§ š‘Ÿ āŠ† (š‘„ Ɨ š‘„)) āˆ§ (š‘Ÿ We š‘„ āˆ§ āˆ€š‘¦ āˆˆ š‘„ (š¹ā€˜(ā—”š‘Ÿ ā€œ {š‘¦})) = š‘¦)))
3029opabbii 5216 . . . 4 {āŸØš‘„, š‘ŸāŸ© āˆ£ ((š‘„ āŠ† š“ āˆ§ š‘Ÿ āŠ† (š‘„ Ɨ š‘„)) āˆ§ (š‘Ÿ We š‘„ āˆ§ āˆ€š‘¦ āˆˆ š‘„ [(ā—”š‘Ÿ ā€œ {š‘¦}) / š‘¢](š‘¢(š¹ āˆ˜ 1st )(š‘Ÿ āˆ© (š‘¢ Ɨ š‘¢))) = š‘¦))} = {āŸØš‘„, š‘ŸāŸ© āˆ£ ((š‘„ āŠ† š“ āˆ§ š‘Ÿ āŠ† (š‘„ Ɨ š‘„)) āˆ§ (š‘Ÿ We š‘„ āˆ§ āˆ€š‘¦ āˆˆ š‘„ (š¹ā€˜(ā—”š‘Ÿ ā€œ {š‘¦})) = š‘¦))}
319, 30eqtr4i 2764 . . 3 š‘Š = {āŸØš‘„, š‘ŸāŸ© āˆ£ ((š‘„ āŠ† š“ āˆ§ š‘Ÿ āŠ† (š‘„ Ɨ š‘„)) āˆ§ (š‘Ÿ We š‘„ āˆ§ āˆ€š‘¦ āˆˆ š‘„ [(ā—”š‘Ÿ ā€œ {š‘¦}) / š‘¢](š‘¢(š¹ āˆ˜ 1st )(š‘Ÿ āˆ© (š‘¢ Ɨ š‘¢))) = š‘¦))}
32 fpwwe.2 . . 3 (šœ‘ ā†’ š“ āˆˆ š‘‰)
33 vex 3479 . . . . 5 š‘„ āˆˆ V
3433, 17opco1i 8111 . . . 4 (š‘„(š¹ āˆ˜ 1st )š‘Ÿ) = (š¹ā€˜š‘„)
35 simp1 1137 . . . . . . 7 ((š‘„ āŠ† š“ āˆ§ š‘Ÿ āŠ† (š‘„ Ɨ š‘„) āˆ§ š‘Ÿ We š‘„) ā†’ š‘„ āŠ† š“)
36 velpw 4608 . . . . . . 7 (š‘„ āˆˆ š’« š“ ā†” š‘„ āŠ† š“)
3735, 36sylibr 233 . . . . . 6 ((š‘„ āŠ† š“ āˆ§ š‘Ÿ āŠ† (š‘„ Ɨ š‘„) āˆ§ š‘Ÿ We š‘„) ā†’ š‘„ āˆˆ š’« š“)
38 19.8a 2175 . . . . . . . 8 (š‘Ÿ We š‘„ ā†’ āˆƒš‘Ÿ š‘Ÿ We š‘„)
39383ad2ant3 1136 . . . . . . 7 ((š‘„ āŠ† š“ āˆ§ š‘Ÿ āŠ† (š‘„ Ɨ š‘„) āˆ§ š‘Ÿ We š‘„) ā†’ āˆƒš‘Ÿ š‘Ÿ We š‘„)
40 ween 10030 . . . . . . 7 (š‘„ āˆˆ dom card ā†” āˆƒš‘Ÿ š‘Ÿ We š‘„)
4139, 40sylibr 233 . . . . . 6 ((š‘„ āŠ† š“ āˆ§ š‘Ÿ āŠ† (š‘„ Ɨ š‘„) āˆ§ š‘Ÿ We š‘„) ā†’ š‘„ āˆˆ dom card)
4237, 41elind 4195 . . . . 5 ((š‘„ āŠ† š“ āˆ§ š‘Ÿ āŠ† (š‘„ Ɨ š‘„) āˆ§ š‘Ÿ We š‘„) ā†’ š‘„ āˆˆ (š’« š“ āˆ© dom card))
43 fpwwe.3 . . . . 5 ((šœ‘ āˆ§ š‘„ āˆˆ (š’« š“ āˆ© dom card)) ā†’ (š¹ā€˜š‘„) āˆˆ š“)
4442, 43sylan2 594 . . . 4 ((šœ‘ āˆ§ (š‘„ āŠ† š“ āˆ§ š‘Ÿ āŠ† (š‘„ Ɨ š‘„) āˆ§ š‘Ÿ We š‘„)) ā†’ (š¹ā€˜š‘„) āˆˆ š“)
4534, 44eqeltrid 2838 . . 3 ((šœ‘ āˆ§ (š‘„ āŠ† š“ āˆ§ š‘Ÿ āŠ† (š‘„ Ɨ š‘„) āˆ§ š‘Ÿ We š‘„)) ā†’ (š‘„(š¹ āˆ˜ 1st )š‘Ÿ) āˆˆ š“)
46 fpwwe.4 . . 3 š‘‹ = āˆŖ dom š‘Š
4731, 32, 45, 46fpwwe2 10638 . 2 (šœ‘ ā†’ ((š‘Œš‘Šš‘… āˆ§ (š‘Œ(š¹ āˆ˜ 1st )š‘…) āˆˆ š‘Œ) ā†” (š‘Œ = š‘‹ āˆ§ š‘… = (š‘Šā€˜š‘‹))))
4816, 47bitr3id 285 1 (šœ‘ ā†’ ((š‘Œš‘Šš‘… āˆ§ (š¹ā€˜š‘Œ) āˆˆ š‘Œ) ā†” (š‘Œ = š‘‹ āˆ§ š‘… = (š‘Šā€˜š‘‹))))
Colors of variables: wff setvar class
Syntax hints:   ā†’ wi 4   ā†” wb 205   āˆ§ wa 397   āˆ§ w3a 1088   = wceq 1542  āˆƒwex 1782   āˆˆ wcel 2107  āˆ€wral 3062  Vcvv 3475  [wsbc 3778   āˆ© cin 3948   āŠ† wss 3949  š’« cpw 4603  {csn 4629  āŸØcop 4635  āˆŖ cuni 4909   class class class wbr 5149  {copab 5211   We wwe 5631   Ɨ cxp 5675  ā—”ccnv 5676  dom cdm 5677   ā€œ cima 5680   āˆ˜ ccom 5681   Fn wfn 6539  ā€“ontoā†’wfo 6542  ā€˜cfv 6544  (class class class)co 7409  1st c1st 7973  cardccrd 9930
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-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  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 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-en 8940  df-oi 9505  df-card 9934
This theorem is referenced by:  canth4  10642  canthnumlem  10643  canthp1lem2  10648
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