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Theorem fpwwe 10587
Description: Given any function š¹ from the powerset of š“ to š“, canth2 9077 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 9971. 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 7361 . . . . . 6 (š‘Œ(š¹ āˆ˜ 1st )š‘…) = ((š¹ āˆ˜ 1st )ā€˜āŸØš‘Œ, š‘…āŸ©)
2 fo1st 7942 . . . . . . . 8 1st :Vā€“ontoā†’V
3 fofn 6759 . . . . . . . 8 (1st :Vā€“ontoā†’V ā†’ 1st Fn V)
42, 3ax-mp 5 . . . . . . 7 1st Fn V
5 opex 5422 . . . . . . 7 āŸØš‘Œ, š‘…āŸ© āˆˆ V
6 fvco2 6939 . . . . . . 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 5724 . . . . . . 7 (š‘Œš‘Šš‘… ā†’ (š‘Œ āˆˆ V āˆ§ š‘… āˆˆ V))
11 op1stg 7934 . . . . . . 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 6847 . . . . 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 3448 . . . . . . . . . . 11 š‘Ÿ āˆˆ V
1817cnvex 7863 . . . . . . . . . 10 ā—”š‘Ÿ āˆˆ V
1918imaex 7854 . . . . . . . . 9 (ā—”š‘Ÿ ā€œ {š‘¦}) āˆˆ V
20 vex 3448 . . . . . . . . . . . 12 š‘¢ āˆˆ V
2117inex1 5275 . . . . . . . . . . . 12 (š‘Ÿ āˆ© (š‘¢ Ɨ š‘¢)) āˆˆ V
2220, 21opco1i 8058 . . . . . . . . . . 11 (š‘¢(š¹ āˆ˜ 1st )(š‘Ÿ āˆ© (š‘¢ Ɨ š‘¢))) = (š¹ā€˜š‘¢)
23 fveq2 6843 . . . . . . . . . . 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 3783 . . . . . . . 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 3093 . . . . . . 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 5173 . . . 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 3448 . . . . 5 š‘„ āˆˆ V
3433, 17opco1i 8058 . . . 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 4566 . . . . . . 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 9976 . . . . . . 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 4155 . . . . 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 10584 . 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 3061  Vcvv 3444  [wsbc 3740   āˆ© cin 3910   āŠ† wss 3911  š’« cpw 4561  {csn 4587  āŸØcop 4593  āˆŖ cuni 4866   class class class wbr 5106  {copab 5168   We wwe 5588   Ɨ cxp 5632  ā—”ccnv 5633  dom cdm 5634   ā€œ cima 5637   āˆ˜ ccom 5638   Fn wfn 6492  ā€“ontoā†’wfo 6495  ā€˜cfv 6497  (class class class)co 7358  1st c1st 7920  cardccrd 9876
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 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
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 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-en 8887  df-oi 9451  df-card 9880
This theorem is referenced by:  canth4  10588  canthnumlem  10589  canthp1lem2  10594
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