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Theorem zfz1isolem1 10804
Description: Lemma for zfz1iso 10805. 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 3273 . . . . 5 |- ( ph -> ( X \ { M } ) C_ ZZ )
3 zfz1isolem1.xf . . . . . 6 |- ( ph -> X e. Fin )
4 zfz1isolem1.mx . . . . . 6 |- ( ph -> M e. X )
5 diffisn 6887 . . . . . 6 |- ( ( X e. Fin /\ M e. X ) -> ( X \ { M } ) e. Fin )
63, 4, 5syl2anc 411 . . . . 5 |- ( ph -> ( X \ { M } ) e. Fin )
7 zfz1isolem1.k . . . . . 6 |- ( ph -> K e. om )
8 zfz1isolem1.xs . . . . . 6 |- ( ph -> X ~~ suc K )
9 dif1en 6873 . . . . . 6 |- ( ( K e. om /\ X ~~ suc K /\ M e. X ) -> ( X \ { M } ) ~~ K )
107, 8, 4, 9syl3anc 1238 . . . . 5 |- ( ph -> ( X \ { M } ) ~~ K )
112, 6, 10jca31 309 . . . 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 3178 . . . . . . . . . 10 |- ( y = ( X \ { M } ) -> ( y C_ ZZ <-> ( X \ { M } ) C_ ZZ ) )
14 eleq1 2240 . . . . . . . . . 10 |- ( y = ( X \ { M } ) -> ( y e. Fin <-> ( X \ { M } ) e. Fin ) )
1513, 14anbi12d 473 . . . . . . . . 9 |- ( y = ( X \ { M } ) -> ( ( y C_ ZZ /\ y e. Fin ) <-> ( ( X \ { M } ) C_ ZZ /\ ( X \ { M } ) e. Fin ) ) )
16 breq1 4003 . . . . . . . . 9 |- ( y = ( X \ { M } ) -> ( y ~~ K <-> ( X \ { M } ) ~~ K ) )
1715, 16anbi12d 473 . . . . . . . 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 5511 . . . . . . . . . . . 12 |- ( y = ( X \ { M } ) -> (♯ ` y ) = (♯ ` ( X \ { M } ) ) )
1918oveq2d 5885 . . . . . . . . . . 11 |- ( y = ( X \ { M } ) -> ( 1 ... (♯ ` y ) ) = ( 1 ... (♯ ` ( X \ { M } ) ) ) )
20 isoeq4 5799 . . . . . . . . . . 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 5800 . . . . . . . . . 10 |- ( y = ( X \ { M } ) -> ( f Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , y ) <-> f Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) )
2321, 22bitrd 188 . . . . . . . . 9 |- ( y = ( X \ { M } ) -> ( f Isom < , < ( ( 1 ... (♯ ` y ) ) , y ) <-> f Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) )
2423exbidv 1825 . . . . . . . 8 |- ( y = ( X \ { M } ) -> ( E. f f Isom < , < ( ( 1 ... (♯ ` y ) ) , y ) <-> E. f f Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) )
2517, 24imbi12d 234 . . . . . . 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 2824 . . . . . 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 5796 . . . 4 |- ( f = g -> ( f Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) <-> g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) )
3130cbvexv 1918 . . 3 |- ( E. f f Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) <-> E. g g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) )
3229, 31sylib 122 . 2 |- ( ph -> E. g g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) )
33 df-isom 5221 . . . . . . . . 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 120 . . . . . . . 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 277 . . . . . . 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 112 . . . . . 6 |- ( ( ph /\ g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) -> g : ( 1 ... (♯ ` ( X \ { M } ) ) ) -1-1-onto-> ( X \ { M } ) )
37 hashcl 10745 . . . . . . . . 9 |- ( X e. Fin -> (♯ ` X ) e. NN0 )
383, 37syl 14 . . . . . . . 8 |- ( ph -> (♯ ` X ) e. NN0 )
3938adantr 276 . . . . . . 7 |- ( ( ph /\ g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) -> (♯ ` X ) e. NN0 )
404adantr 276 . . . . . . 7 |- ( ( ph /\ g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) -> M e. X )
41 f1osng 5498 . . . . . . 7 |- ( ( (♯ ` X ) e. NN0 /\ M e. X ) -> { <. (♯ ` X ) , M >. } : { (♯ ` X ) } -1-1-onto-> { M } )
4239, 40, 41syl2anc 411 . . . . . 6 |- ( ( ph /\ g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) -> { <. (♯ ` X ) , M >. } : { (♯ ` X ) } -1-1-onto-> { M } )
43 hashdifsn 10783 . . . . . . . . . . . . 13 |- ( ( X e. Fin /\ M e. X ) -> (♯ ` ( X \ { M } ) ) = ( (♯ ` X ) - 1 ) )
443, 4, 43syl2anc 411 . . . . . . . . . . . 12 |- ( ph -> (♯ ` ( X \ { M } ) ) = ( (♯ ` X ) - 1 ) )
4544oveq1d 5884 . . . . . . . . . . 11 |- ( ph -> ( (♯ ` ( X \ { M } ) ) + 1 ) = ( ( (♯ ` X ) - 1 ) + 1 ) )
4638nn0cnd 9220 . . . . . . . . . . . 12 |- ( ph -> (♯ ` X ) e. CC )
47 1cnd 7964 . . . . . . . . . . . 12 |- ( ph -> 1 e. CC )
4846, 47npcand 8262 . . . . . . . . . . 11 |- ( ph -> ( ( (♯ ` X ) - 1 ) + 1 ) = (♯ ` X ) )
4945, 48eqtrd 2210 . . . . . . . . . 10 |- ( ph -> ( (♯ ` ( X \ { M } ) ) + 1 ) = (♯ ` X ) )
5049sneqd 3604 . . . . . . . . 9 |- ( ph -> { ( (♯ ` ( X \ { M } ) ) + 1 ) } = { (♯ ` X ) } )
5150ineq2d 3336 . . . . . . . 8 |- ( ph -> ( ( 1 ... (♯ ` ( X \ { M } ) ) ) i^i { ( (♯ ` ( X \ { M } ) ) + 1 ) } ) = ( ( 1 ... (♯ ` ( X \ { M } ) ) ) i^i { (♯ ` X ) } ) )
52 fzp1disj 10066 . . . . . . . 8 |- ( ( 1 ... (♯ ` ( X \ { M } ) ) ) i^i { ( (♯ ` ( X \ { M } ) ) + 1 ) } ) = (/)
5351, 52eqtr3di 2225 . . . . . . 7 |- ( ph -> ( ( 1 ... (♯ ` ( X \ { M } ) ) ) i^i { (♯ ` X ) } ) = (/) )
5453adantr 276 . . . . . 6 |- ( ( ph /\ g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) -> ( ( 1 ... (♯ ` ( X \ { M } ) ) ) i^i { (♯ ` X ) } ) = (/) )
55 incom 3327 . . . . . . . 8 |- ( ( X \ { M } ) i^i { M } ) = ( { M } i^i ( X \ { M } ) )
56 disjdif 3495 . . . . . . . 8 |- ( { M } i^i ( X \ { M } ) ) = (/)
5755, 56eqtri 2198 . . . . . . 7 |- ( ( X \ { M } ) i^i { M } ) = (/)
5857a1i 9 . . . . . 6 |- ( ( ph /\ g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) -> ( ( X \ { M } ) i^i { M } ) = (/) )
59 f1oun 5477 . . . . . 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 1239 . . . . 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 10802 . . . . . . 7 |- ( ph -> ( 1 ... (♯ ` X ) ) = ( ( 1 ... (♯ ` ( X \ { M } ) ) ) u. { (♯ ` X ) } ) )
62 fidifsnid 6865 . . . . . . . . 9 |- ( ( X e. Fin /\ M e. X ) -> ( ( X \ { M } ) u. { M } ) = X )
633, 4, 62syl2anc 411 . . . . . . . 8 |- ( ph -> ( ( X \ { M } ) u. { M } ) = X )
6463eqcomd 2183 . . . . . . 7 |- ( ph -> X = ( ( X \ { M } ) u. { M } ) )
65 f1oeq23 5448 . . . . . . 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 411 . . . . . 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 276 . . . . 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 167 . . . 4 |- ( ( ph /\ g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) -> ( g u. { <. (♯ ` X ) , M >. } ) : ( 1 ... (♯ ` X ) ) -1-1-onto-> X )
693ad2antrr 488 . . . . . 6 |- ( ( ( ph /\ g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) /\ ( a e. ( 1 ... (♯ ` X ) ) /\ b e. ( 1 ... (♯ ` X ) ) ) ) -> X e. Fin )
701ad2antrr 488 . . . . . 6 |- ( ( ( ph /\ g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) /\ ( a e. ( 1 ... (♯ ` X ) ) /\ b e. ( 1 ... (♯ ` X ) ) ) ) -> X C_ ZZ )
714ad2antrr 488 . . . . . 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 488 . . . . . 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 528 . . . . . 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 529 . . . . . 6 |- ( ( ( ph /\ g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) /\ ( a e. ( 1 ... (♯ ` X ) ) /\ b e. ( 1 ... (♯ ` X ) ) ) ) -> a e. ( 1 ... (♯ ` X ) ) )
76 simprr 531 . . . . . 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 10803 . . . . 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 2559 . . . 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 5221 . . . 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 417 . . 3 |- ( ( ph /\ g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) -> ( g u. { <. (♯ ` X ) , M >. } ) Isom < , < ( ( 1 ... (♯ ` X ) ) , X ) )
81 vex 2740 . . . . . . 7 |- g e. _V
8281a1i 9 . . . . . 6 |- ( ph -> g e. _V )
83 opexg 4225 . . . . . . . 8 |- ( ( (♯ ` X ) e. NN0 /\ M e. X ) -> <. (♯ ` X ) , M >. e. _V )
8438, 4, 83syl2anc 411 . . . . . . 7 |- ( ph -> <. (♯ ` X ) , M >. e. _V )
85 snexg 4181 . . . . . . 7 |- ( <. (♯ ` X ) , M >. e. _V -> { <. (♯ ` X ) , M >. } e. _V )
8684, 85syl 14 . . . . . 6 |- ( ph -> { <. (♯ ` X ) , M >. } e. _V )
87 unexg 4440 . . . . . 6 |- ( ( g e. _V /\ { <. (♯ ` X ) , M >. } e. _V ) -> ( g u. { <. (♯ ` X ) , M >. } ) e. _V )
8882, 86, 87syl2anc 411 . . . . 5 |- ( ph -> ( g u. { <. (♯ ` X ) , M >. } ) e. _V )
89 isoeq1 5796 . . . . . 6 |- ( f = ( g u. { <. (♯ ` X ) , M >. } ) -> ( f Isom < , < ( ( 1 ... (♯ ` X ) ) , X ) <-> ( g u. { <. (♯ ` X ) , M >. } ) Isom < , < ( ( 1 ... (♯ ` X ) ) , X ) ) )
9089spcegv 2825 . . . . 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 276 . . 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 1898 1 |- ( ph -> E. f f Isom < , < ( ( 1 ... (♯ ` X ) ) , X ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1351   = wceq 1353   E.wex 1492   e. wcel 2148   A.wral 2455   _Vcvv 2737   \ cdif 3126   u. cun 3127   i^i cin 3128   C_ wss 3129   (/)c0 3422   {csn 3591   <.cop 3594   class class class wbr 4000   suc csuc 4362   omcom 4586   -1-1-onto->wf1o 5211   ` cfv 5212   Isom wiso 5213  (class class class)co 5869   ~~ cen 6732   Fincfn 6734   1c1 7803   + caddc 7805   < clt 7982   <_ cle 7983   - cmin 8118   NN0cn0 9165   ZZcz 9242   ...cfz 9995  ♯chash 10739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-0id 7910  ax-rnegex 7911  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-frec 6386  df-1o 6411  df-oadd 6415  df-er 6529  df-en 6735  df-dom 6736  df-fin 6737  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-inn 8909  df-n0 9166  df-z 9243  df-uz 9518  df-fz 9996  df-ihash 10740
This theorem is referenced by:  zfz1iso  10805
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