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Theorem zfz1isolem1 10775
Description: Lemma for zfz1iso 10776. Existence of an order isomorphism given the existence of shorter isomorphisms. (Contributed by Jim Kingdon, 7-Sep-2022.)
Hypotheses
Ref Expression
zfz1isolem1.k |- ( ph -> K e. om )
zfz1isolem1.h |- ( ph -> A. y ( ( ( y C_ ZZ /\ y e. Fin ) /\ y ~~ K ) -> E. f f Isom < , < ( ( 1 ... (♯ ` y ) ) , y ) ) )
zfz1isolem1.xz |- ( ph -> X C_ ZZ )
zfz1isolem1.xf |- ( ph -> X e. Fin )
zfz1isolem1.xs |- ( ph -> X ~~ suc K )
zfz1isolem1.mx |- ( ph -> M e. X )
zfz1isolem1.m |- ( ph -> A. z e. X z <_ M )
Assertion
Ref Expression
zfz1isolem1 |- ( ph -> E. f f Isom < , < ( ( 1 ... (♯ ` X ) ) , X ) )
Distinct variable groups:   y, K   z, M   f, M, y   z, X   f, X, y
Allowed substitution hints:   ph( y, z, f)   K( z, f)
Proof of Theorem zfz1isolem1
Dummy variables a b g u v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 zfz1isolem1.xz . . . . . 6 |- ( ph -> X C_ ZZ )
21ssdifssd 3265 . . . . 5 |- ( ph -> ( X \ { M } ) C_ ZZ )
3 zfz1isolem1.xf . . . . . 6 |- ( ph -> X e. Fin )
4 zfz1isolem1.mx . . . . . 6 |- ( ph -> M e. X )
5 diffisn 6871 . . . . . 6 |- ( ( X e. Fin /\ M e. X ) -> ( X \ { M } ) e. Fin )
63, 4, 5syl2anc 409 . . . . 5 |- ( ph -> ( X \ { M } ) e. Fin )
7 zfz1isolem1.k . . . . . 6 |- ( ph -> K e. om )
8 zfz1isolem1.xs . . . . . 6 |- ( ph -> X ~~ suc K )
9 dif1en 6857 . . . . . 6 |- ( ( K e. om /\ X ~~ suc K /\ M e. X ) -> ( X \ { M } ) ~~ K )
107, 8, 4, 9syl3anc 1233 . . . . 5 |- ( ph -> ( X \ { M } ) ~~ K )
112, 6, 10jca31 307 . . . 4 |- ( ph -> ( ( ( X \ { M } ) C_ ZZ /\ ( X \ { M } ) e. Fin ) /\ ( X \ { M } ) ~~ K ) )
12 zfz1isolem1.h . . . . 5 |- ( ph -> A. y ( ( ( y C_ ZZ /\ y e. Fin ) /\ y ~~ K ) -> E. f f Isom < , < ( ( 1 ... (♯ ` y ) ) , y ) ) )
13 sseq1 3170 . . . . . . . . . 10 |- ( y = ( X \ { M } ) -> ( y C_ ZZ <-> ( X \ { M } ) C_ ZZ ) )
14 eleq1 2233 . . . . . . . . . 10 |- ( y = ( X \ { M } ) -> ( y e. Fin <-> ( X \ { M } ) e. Fin ) )
1513, 14anbi12d 470 . . . . . . . . 9 |- ( y = ( X \ { M } ) -> ( ( y C_ ZZ /\ y e. Fin ) <-> ( ( X \ { M } ) C_ ZZ /\ ( X \ { M } ) e. Fin ) ) )
16 breq1 3992 . . . . . . . . 9 |- ( y = ( X \ { M } ) -> ( y ~~ K <-> ( X \ { M } ) ~~ K ) )
1715, 16anbi12d 470 . . . . . . . 8 |- ( y = ( X \ { M } ) -> ( ( ( y C_ ZZ /\ y e. Fin ) /\ y ~~ K ) <-> ( ( ( X \ { M } ) C_ ZZ /\ ( X \ { M } ) e. Fin ) /\ ( X \ { M } ) ~~ K ) ) )
18 fveq2 5496 . . . . . . . . . . . 12 |- ( y = ( X \ { M } ) -> (♯ ` y ) = (♯ ` ( X \ { M } ) ) )
1918oveq2d 5869 . . . . . . . . . . 11 |- ( y = ( X \ { M } ) -> ( 1 ... (♯ ` y ) ) = ( 1 ... (♯ ` ( X \ { M } ) ) ) )
20 isoeq4 5783 . . . . . . . . . . 11 |- ( ( 1 ... (♯ ` y ) ) = ( 1 ... (♯ ` ( X \ { M } ) ) ) -> ( f Isom < , < ( ( 1 ... (♯ ` y ) ) , y ) <-> f Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , y ) ) )
2119, 20syl 14 . . . . . . . . . 10 |- ( y = ( X \ { M } ) -> ( f Isom < , < ( ( 1 ... (♯ ` y ) ) , y ) <-> f Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , y ) ) )
22 isoeq5 5784 . . . . . . . . . 10 |- ( y = ( X \ { M } ) -> ( f Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , y ) <-> f Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) )
2321, 22bitrd 187 . . . . . . . . 9 |- ( y = ( X \ { M } ) -> ( f Isom < , < ( ( 1 ... (♯ ` y ) ) , y ) <-> f Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) )
2423exbidv 1818 . . . . . . . 8 |- ( y = ( X \ { M } ) -> ( E. f f Isom < , < ( ( 1 ... (♯ ` y ) ) , y ) <-> E. f f Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) )
2517, 24imbi12d 233 . . . . . . 7 |- ( y = ( X \ { M } ) -> ( ( ( ( y C_ ZZ /\ y e. Fin ) /\ y ~~ K ) -> E. f f Isom < , < ( ( 1 ... (♯ ` y ) ) , y ) ) <-> ( ( ( ( X \ { M } ) C_ ZZ /\ ( X \ { M } ) e. Fin ) /\ ( X \ { M } ) ~~ K ) -> E. f f Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) ) )
2625spcgv 2817 . . . . . 6 |- ( ( X \ { M } ) e. Fin -> ( A. y ( ( ( y C_ ZZ /\ y e. Fin ) /\ y ~~ K ) -> E. f f Isom < , < ( ( 1 ... (♯ ` y ) ) , y ) ) -> ( ( ( ( X \ { M } ) C_ ZZ /\ ( X \ { M } ) e. Fin ) /\ ( X \ { M } ) ~~ K ) -> E. f f Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) ) )
276, 26syl 14 . . . . 5 |- ( ph -> ( A. y ( ( ( y C_ ZZ /\ y e. Fin ) /\ y ~~ K ) -> E. f f Isom < , < ( ( 1 ... (♯ ` y ) ) , y ) ) -> ( ( ( ( X \ { M } ) C_ ZZ /\ ( X \ { M } ) e. Fin ) /\ ( X \ { M } ) ~~ K ) -> E. f f Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) ) )
2812, 27mpd 13 . . . 4 |- ( ph -> ( ( ( ( X \ { M } ) C_ ZZ /\ ( X \ { M } ) e. Fin ) /\ ( X \ { M } ) ~~ K ) -> E. f f Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) )
2911, 28mpd 13 . . 3 |- ( ph -> E. f f Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) )
30 isoeq1 5780 . . . 4 |- ( f = g -> ( f Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) <-> g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) )
3130cbvexv 1911 . . 3 |- ( E. f f Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) <-> E. g g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) )
3229, 31sylib 121 . 2 |- ( ph -> E. g g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) )
33 df-isom 5207 . . . . . . . . 9 |- ( g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) <-> ( g : ( 1 ... (♯ ` ( X \ { M } ) ) ) -1-1-onto-> ( X \ { M } ) /\ A. u e. ( 1 ... (♯ ` ( X \ { M } ) ) ) A. v e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ( u < v <-> ( g ` u ) < ( g ` v ) ) ) )
3433biimpi 119 . . . . . . . 8 |- ( g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) -> ( g : ( 1 ... (♯ ` ( X \ { M } ) ) ) -1-1-onto-> ( X \ { M } ) /\ A. u e. ( 1 ... (♯ ` ( X \ { M } ) ) ) A. v e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ( u < v <-> ( g ` u ) < ( g ` v ) ) ) )
3534adantl 275 . . . . . . 7 |- ( ( ph /\ g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) -> ( g : ( 1 ... (♯ ` ( X \ { M } ) ) ) -1-1-onto-> ( X \ { M } ) /\ A. u e. ( 1 ... (♯ ` ( X \ { M } ) ) ) A. v e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ( u < v <-> ( g ` u ) < ( g ` v ) ) ) )
3635simpld 111 . . . . . 6 |- ( ( ph /\ g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) -> g : ( 1 ... (♯ ` ( X \ { M } ) ) ) -1-1-onto-> ( X \ { M } ) )
37 hashcl 10715 . . . . . . . . 9 |- ( X e. Fin -> (♯ ` X ) e. NN0 )
383, 37syl 14 . . . . . . . 8 |- ( ph -> (♯ ` X ) e. NN0 )
3938adantr 274 . . . . . . 7 |- ( ( ph /\ g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) -> (♯ ` X ) e. NN0 )
404adantr 274 . . . . . . 7 |- ( ( ph /\ g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) -> M e. X )
41 f1osng 5483 . . . . . . 7 |- ( ( (♯ ` X ) e. NN0 /\ M e. X ) -> { <. (♯ ` X ) , M >. } : { (♯ ` X ) } -1-1-onto-> { M } )
4239, 40, 41syl2anc 409 . . . . . 6 |- ( ( ph /\ g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) -> { <. (♯ ` X ) , M >. } : { (♯ ` X ) } -1-1-onto-> { M } )
43 hashdifsn 10754 . . . . . . . . . . . . 13 |- ( ( X e. Fin /\ M e. X ) -> (♯ ` ( X \ { M } ) ) = ( (♯ ` X ) - 1 ) )
443, 4, 43syl2anc 409 . . . . . . . . . . . 12 |- ( ph -> (♯ ` ( X \ { M } ) ) = ( (♯ ` X ) - 1 ) )
4544oveq1d 5868 . . . . . . . . . . 11 |- ( ph -> ( (♯ ` ( X \ { M } ) ) + 1 ) = ( ( (♯ ` X ) - 1 ) + 1 ) )
4638nn0cnd 9190 . . . . . . . . . . . 12 |- ( ph -> (♯ ` X ) e. CC )
47 1cnd 7936 . . . . . . . . . . . 12 |- ( ph -> 1 e. CC )
4846, 47npcand 8234 . . . . . . . . . . 11 |- ( ph -> ( ( (♯ ` X ) - 1 ) + 1 ) = (♯ ` X ) )
4945, 48eqtrd 2203 . . . . . . . . . 10 |- ( ph -> ( (♯ ` ( X \ { M } ) ) + 1 ) = (♯ ` X ) )
5049sneqd 3596 . . . . . . . . 9 |- ( ph -> { ( (♯ ` ( X \ { M } ) ) + 1 ) } = { (♯ ` X ) } )
5150ineq2d 3328 . . . . . . . 8 |- ( ph -> ( ( 1 ... (♯ ` ( X \ { M } ) ) ) i^i { ( (♯ ` ( X \ { M } ) ) + 1 ) } ) = ( ( 1 ... (♯ ` ( X \ { M } ) ) ) i^i { (♯ ` X ) } ) )
52 fzp1disj 10036 . . . . . . . 8 |- ( ( 1 ... (♯ ` ( X \ { M } ) ) ) i^i { ( (♯ ` ( X \ { M } ) ) + 1 ) } ) = (/)
5351, 52eqtr3di 2218 . . . . . . 7 |- ( ph -> ( ( 1 ... (♯ ` ( X \ { M } ) ) ) i^i { (♯ ` X ) } ) = (/) )
5453adantr 274 . . . . . 6 |- ( ( ph /\ g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) -> ( ( 1 ... (♯ ` ( X \ { M } ) ) ) i^i { (♯ ` X ) } ) = (/) )
55 incom 3319 . . . . . . . 8 |- ( ( X \ { M } ) i^i { M } ) = ( { M } i^i ( X \ { M } ) )
56 disjdif 3487 . . . . . . . 8 |- ( { M } i^i ( X \ { M } ) ) = (/)
5755, 56eqtri 2191 . . . . . . 7 |- ( ( X \ { M } ) i^i { M } ) = (/)
5857a1i 9 . . . . . 6 |- ( ( ph /\ g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) -> ( ( X \ { M } ) i^i { M } ) = (/) )
59 f1oun 5462 . . . . . 6 |- ( ( ( g : ( 1 ... (♯ ` ( X \ { M } ) ) ) -1-1-onto-> ( X \ { M } ) /\ { <. (♯ ` X ) , M >. } : { (♯ ` X ) } -1-1-onto-> { M } ) /\ ( ( ( 1 ... (♯ ` ( X \ { M } ) ) ) i^i { (♯ ` X ) } ) = (/) /\ ( ( X \ { M } ) i^i { M } ) = (/) ) ) -> ( g u. { <. (♯ ` X ) , M >. } ) : ( ( 1 ... (♯ ` ( X \ { M } ) ) ) u. { (♯ ` X ) } ) -1-1-onto-> ( ( X \ { M } ) u. { M } ) )
6036, 42, 54, 58, 59syl22anc 1234 . . . . 5 |- ( ( ph /\ g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) -> ( g u. { <. (♯ ` X ) , M >. } ) : ( ( 1 ... (♯ ` ( X \ { M } ) ) ) u. { (♯ ` X ) } ) -1-1-onto-> ( ( X \ { M } ) u. { M } ) )
613, 4zfz1isolemsplit 10773 . . . . . . 7 |- ( ph -> ( 1 ... (♯ ` X ) ) = ( ( 1 ... (♯ ` ( X \ { M } ) ) ) u. { (♯ ` X ) } ) )
62 fidifsnid 6849 . . . . . . . . 9 |- ( ( X e. Fin /\ M e. X ) -> ( ( X \ { M } ) u. { M } ) = X )
633, 4, 62syl2anc 409 . . . . . . . 8 |- ( ph -> ( ( X \ { M } ) u. { M } ) = X )
6463eqcomd 2176 . . . . . . 7 |- ( ph -> X = ( ( X \ { M } ) u. { M } ) )
65 f1oeq23 5434 . . . . . . 7 |- ( ( ( 1 ... (♯ ` X ) ) = ( ( 1 ... (♯ ` ( X \ { M } ) ) ) u. { (♯ ` X ) } ) /\ X = ( ( X \ { M } ) u. { M } ) ) -> ( ( g u. { <. (♯ ` X ) , M >. } ) : ( 1 ... (♯ ` X ) ) -1-1-onto-> X <-> ( g u. { <. (♯ ` X ) , M >. } ) : ( ( 1 ... (♯ ` ( X \ { M } ) ) ) u. { (♯ ` X ) } ) -1-1-onto-> ( ( X \ { M } ) u. { M } ) ) )
6661, 64, 65syl2anc 409 . . . . . 6 |- ( ph -> ( ( g u. { <. (♯ ` X ) , M >. } ) : ( 1 ... (♯ ` X ) ) -1-1-onto-> X <-> ( g u. { <. (♯ ` X ) , M >. } ) : ( ( 1 ... (♯ ` ( X \ { M } ) ) ) u. { (♯ ` X ) } ) -1-1-onto-> ( ( X \ { M } ) u. { M } ) ) )
6766adantr 274 . . . . 5 |- ( ( ph /\ g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) -> ( ( g u. { <. (♯ ` X ) , M >. } ) : ( 1 ... (♯ ` X ) ) -1-1-onto-> X <-> ( g u. { <. (♯ ` X ) , M >. } ) : ( ( 1 ... (♯ ` ( X \ { M } ) ) ) u. { (♯ ` X ) } ) -1-1-onto-> ( ( X \ { M } ) u. { M } ) ) )
6860, 67mpbird 166 . . . 4 |- ( ( ph /\ g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) -> ( g u. { <. (♯ ` X ) , M >. } ) : ( 1 ... (♯ ` X ) ) -1-1-onto-> X )
693ad2antrr 485 . . . . . 6 |- ( ( ( ph /\ g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) /\ ( a e. ( 1 ... (♯ ` X ) ) /\ b e. ( 1 ... (♯ ` X ) ) ) ) -> X e. Fin )
701ad2antrr 485 . . . . . 6 |- ( ( ( ph /\ g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) /\ ( a e. ( 1 ... (♯ ` X ) ) /\ b e. ( 1 ... (♯ ` X ) ) ) ) -> X C_ ZZ )
714ad2antrr 485 . . . . . 6 |- ( ( ( ph /\ g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) /\ ( a e. ( 1 ... (♯ ` X ) ) /\ b e. ( 1 ... (♯ ` X ) ) ) ) -> M e. X )
72 zfz1isolem1.m . . . . . . 7 |- ( ph -> A. z e. X z <_ M )
7372ad2antrr 485 . . . . . 6 |- ( ( ( ph /\ g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) /\ ( a e. ( 1 ... (♯ ` X ) ) /\ b e. ( 1 ... (♯ ` X ) ) ) ) -> A. z e. X z <_ M )
74 simplr 525 . . . . . 6 |- ( ( ( ph /\ g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) /\ ( a e. ( 1 ... (♯ ` X ) ) /\ b e. ( 1 ... (♯ ` X ) ) ) ) -> g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) )
75 simprl 526 . . . . . 6 |- ( ( ( ph /\ g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) /\ ( a e. ( 1 ... (♯ ` X ) ) /\ b e. ( 1 ... (♯ ` X ) ) ) ) -> a e. ( 1 ... (♯ ` X ) ) )
76 simprr 527 . . . . . 6 |- ( ( ( ph /\ g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) /\ ( a e. ( 1 ... (♯ ` X ) ) /\ b e. ( 1 ... (♯ ` X ) ) ) ) -> b e. ( 1 ... (♯ ` X ) ) )
7769, 70, 71, 73, 74, 75, 76zfz1isolemiso 10774 . . . . 5 |- ( ( ( ph /\ g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) /\ ( a e. ( 1 ... (♯ ` X ) ) /\ b e. ( 1 ... (♯ ` X ) ) ) ) -> ( a < b <-> ( ( g u. { <. (♯ ` X ) , M >. } ) ` a ) < ( ( g u. { <. (♯ ` X ) , M >. } ) ` b ) ) )
7877ralrimivva 2552 . . . 4 |- ( ( ph /\ g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) -> A. a e. ( 1 ... (♯ ` X ) ) A. b e. ( 1 ... (♯ ` X ) ) ( a < b <-> ( ( g u. { <. (♯ ` X ) , M >. } ) ` a ) < ( ( g u. { <. (♯ ` X ) , M >. } ) ` b ) ) )
79 df-isom 5207 . . . 4 |- ( ( g u. { <. (♯ ` X ) , M >. } ) Isom < , < ( ( 1 ... (♯ ` X ) ) , X ) <-> ( ( g u. { <. (♯ ` X ) , M >. } ) : ( 1 ... (♯ ` X ) ) -1-1-onto-> X /\ A. a e. ( 1 ... (♯ ` X ) ) A. b e. ( 1 ... (♯ ` X ) ) ( a < b <-> ( ( g u. { <. (♯ ` X ) , M >. } ) ` a ) < ( ( g u. { <. (♯ ` X ) , M >. } ) ` b ) ) ) )
8068, 78, 79sylanbrc 415 . . 3 |- ( ( ph /\ g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) -> ( g u. { <. (♯ ` X ) , M >. } ) Isom < , < ( ( 1 ... (♯ ` X ) ) , X ) )
81 vex 2733 . . . . . . 7 |- g e. _V
8281a1i 9 . . . . . 6 |- ( ph -> g e. _V )
83 opexg 4213 . . . . . . . 8 |- ( ( (♯ ` X ) e. NN0 /\ M e. X ) -> <. (♯ ` X ) , M >. e. _V )
8438, 4, 83syl2anc 409 . . . . . . 7 |- ( ph -> <. (♯ ` X ) , M >. e. _V )
85 snexg 4170 . . . . . . 7 |- ( <. (♯ ` X ) , M >. e. _V -> { <. (♯ ` X ) , M >. } e. _V )
8684, 85syl 14 . . . . . 6 |- ( ph -> { <. (♯ ` X ) , M >. } e. _V )
87 unexg 4428 . . . . . 6 |- ( ( g e. _V /\ { <. (♯ ` X ) , M >. } e. _V ) -> ( g u. { <. (♯ ` X ) , M >. } ) e. _V )
8882, 86, 87syl2anc 409 . . . . 5 |- ( ph -> ( g u. { <. (♯ ` X ) , M >. } ) e. _V )
89 isoeq1 5780 . . . . . 6 |- ( f = ( g u. { <. (♯ ` X ) , M >. } ) -> ( f Isom < , < ( ( 1 ... (♯ ` X ) ) , X ) <-> ( g u. { <. (♯ ` X ) , M >. } ) Isom < , < ( ( 1 ... (♯ ` X ) ) , X ) ) )
9089spcegv 2818 . . . . 5 |- ( ( g u. { <. (♯ ` X ) , M >. } ) e. _V -> ( ( g u. { <. (♯ ` X ) , M >. } ) Isom < , < ( ( 1 ... (♯ ` X ) ) , X ) -> E. f f Isom < , < ( ( 1 ... (♯ ` X ) ) , X ) ) )
9188, 90syl 14 . . . 4 |- ( ph -> ( ( g u. { <. (♯ ` X ) , M >. } ) Isom < , < ( ( 1 ... (♯ ` X ) ) , X ) -> E. f f Isom < , < ( ( 1 ... (♯ ` X ) ) , X ) ) )
9291adantr 274 . . 3 |- ( ( ph /\ g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) -> ( ( g u. { <. (♯ ` X ) , M >. } ) Isom < , < ( ( 1 ... (♯ ` X ) ) , X ) -> E. f f Isom < , < ( ( 1 ... (♯ ` X ) ) , X ) ) )
9380, 92mpd 13 . 2 |- ( ( ph /\ g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) -> E. f f Isom < , < ( ( 1 ... (♯ ` X ) ) , X ) )
9432, 93exlimddv 1891 1 |- ( ph -> E. f f Isom < , < ( ( 1 ... (♯ ` X ) ) , X ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1346   = wceq 1348   E.wex 1485   e. wcel 2141   A.wral 2448   _Vcvv 2730   \ cdif 3118   u. cun 3119   i^i cin 3120   C_ wss 3121   (/)c0 3414   {csn 3583   <.cop 3586   class class class wbr 3989   suc csuc 4350   omcom 4574   -1-1-onto->wf1o 5197   ` cfv 5198   Isom wiso 5199  (class class class)co 5853   ~~ cen 6716   Fincfn 6718   1c1 7775   + caddc 7777   < clt 7954   <_ cle 7955   - cmin 8090   NN0cn0 9135   ZZcz 9212   ...cfz 9965  ♯chash 10709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-frec 6370  df-1o 6395  df-oadd 6399  df-er 6513  df-en 6719  df-dom 6720  df-fin 6721  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488  df-fz 9966  df-ihash 10710
This theorem is referenced by:  zfz1iso  10776
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