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Theorem zfz1isolem1 11270
Description: Lemma for zfz1iso 11271. 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 3367 . . . . 5 |- ( ph -> ( X \ { M } ) C_ ZZ )
3 zfz1isolem1.xf . . . . . 6 |- ( ph -> X e. Fin )
4 zfz1isolem1.mx . . . . . 6 |- ( ph -> M e. X )
5 diffisn 7187 . . . . . 6 |- ( ( X e. Fin /\ M e. X ) -> ( X \ { M } ) e. Fin )
63, 4, 5syl2anc 415 . . . . 5 |- ( ph -> ( X \ { M } ) e. Fin )
7 zfz1isolem1.k . . . . . 6 |- ( ph -> K e. om )
8 zfz1isolem1.xs . . . . . 6 |- ( ph -> X ~~ suc K )
9 dif1en 7173 . . . . . 6 |- ( ( K e. om /\ X ~~ suc K /\ M e. X ) -> ( X \ { M } ) ~~ K )
107, 8, 4, 9syl3anc 1278 . . . . 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 3271 . . . . . . . . . 10 |- ( y = ( X \ { M } ) -> ( y C_ ZZ <-> ( X \ { M } ) C_ ZZ ) )
14 eleq1 2301 . . . . . . . . . 10 |- ( y = ( X \ { M } ) -> ( y e. Fin <-> ( X \ { M } ) e. Fin ) )
1513, 14anbi12d 477 . . . . . . . . 9 |- ( y = ( X \ { M } ) -> ( ( y C_ ZZ /\ y e. Fin ) <-> ( ( X \ { M } ) C_ ZZ /\ ( X \ { M } ) e. Fin ) ) )
16 breq1 4128 . . . . . . . . 9 |- ( y = ( X \ { M } ) -> ( y ~~ K <-> ( X \ { M } ) ~~ K ) )
1715, 16anbi12d 477 . . . . . . . 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 5690 . . . . . . . . . . . 12 |- ( y = ( X \ { M } ) -> (♯ ` y ) = (♯ ` ( X \ { M } ) ) )
1918oveq2d 6091 . . . . . . . . . . 11 |- ( y = ( X \ { M } ) -> ( 1 ... (♯ ` y ) ) = ( 1 ... (♯ ` ( X \ { M } ) ) ) )
20 isoeq4 6000 . . . . . . . . . . 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 6001 . . . . . . . . . 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 1878 . . . . . . . 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 2912 . . . . . 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 5997 . . . 4 |- ( f = g -> ( f Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) <-> g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) )
3130cbvexv 1974 . . 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 5381 . . . . . . . . 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 11198 . . . . . . . . 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 5677 . . . . . . 7 |- ( ( (♯ ` X ) e. NN0 /\ M e. X ) -> { <. (♯ ` X ) , M >. } : { (♯ ` X ) } -1-1-onto-> { M } )
4239, 40, 41syl2anc 415 . . . . . 6 |- ( ( ph /\ g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) -> { <. (♯ ` X ) , M >. } : { (♯ ` X ) } -1-1-onto-> { M } )
43 hashdifsn 11238 . . . . . . . . . . . . 13 |- ( ( X e. Fin /\ M e. X ) -> (♯ ` ( X \ { M } ) ) = ( (♯ ` X ) - 1 ) )
443, 4, 43syl2anc 415 . . . . . . . . . . . 12 |- ( ph -> (♯ ` ( X \ { M } ) ) = ( (♯ ` X ) - 1 ) )
4544oveq1d 6090 . . . . . . . . . . 11 |- ( ph -> ( (♯ ` ( X \ { M } ) ) + 1 ) = ( ( (♯ ` X ) - 1 ) + 1 ) )
4638nn0cnd 9601 . . . . . . . . . . . 12 |- ( ph -> (♯ ` X ) e. CC )
47 1cnd 8332 . . . . . . . . . . . 12 |- ( ph -> 1 e. CC )
4846, 47npcand 8631 . . . . . . . . . . 11 |- ( ph -> ( ( (♯ ` X ) - 1 ) + 1 ) = (♯ ` X ) )
4945, 48eqtrd 2271 . . . . . . . . . 10 |- ( ph -> ( (♯ ` ( X \ { M } ) ) + 1 ) = (♯ ` X ) )
5049sneqd 3718 . . . . . . . . 9 |- ( ph -> { ( (♯ ` ( X \ { M } ) ) + 1 ) } = { (♯ ` X ) } )
5150ineq2d 3432 . . . . . . . 8 |- ( ph -> ( ( 1 ... (♯ ` ( X \ { M } ) ) ) i^i { ( (♯ ` ( X \ { M } ) ) + 1 ) } ) = ( ( 1 ... (♯ ` ( X \ { M } ) ) ) i^i { (♯ ` X ) } ) )
52 fzp1disj 10465 . . . . . . . 8 |- ( ( 1 ... (♯ ` ( X \ { M } ) ) ) i^i { ( (♯ ` ( X \ { M } ) ) + 1 ) } ) = (/)
5351, 52eqtr3di 2286 . . . . . . 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 3421 . . . . . . . 8 |- ( ( X \ { M } ) i^i { M } ) = ( { M } i^i ( X \ { M } ) )
56 disjdif 3596 . . . . . . . 8 |- ( { M } i^i ( X \ { M } ) ) = (/)
5755, 56eqtri 2259 . . . . . . 7 |- ( ( X \ { M } ) i^i { M } ) = (/)
5857a1i 9 . . . . . 6 |- ( ( ph /\ g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) -> ( ( X \ { M } ) i^i { M } ) = (/) )
59 f1oun 5654 . . . . . 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 1279 . . . . 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 11268 . . . . . . 7 |- ( ph -> ( 1 ... (♯ ` X ) ) = ( ( 1 ... (♯ ` ( X \ { M } ) ) ) u. { (♯ ` X ) } ) )
62 fidifsnid 7163 . . . . . . . . 9 |- ( ( X e. Fin /\ M e. X ) -> ( ( X \ { M } ) u. { M } ) = X )
633, 4, 62syl2anc 415 . . . . . . . 8 |- ( ph -> ( ( X \ { M } ) u. { M } ) = X )
6463eqcomd 2244 . . . . . . 7 |- ( ph -> X = ( ( X \ { M } ) u. { M } ) )
65 f1oeq23 5625 . . . . . . 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 415 . . . . . 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 492 . . . . . 6 |- ( ( ( ph /\ g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) /\ ( a e. ( 1 ... (♯ ` X ) ) /\ b e. ( 1 ... (♯ ` X ) ) ) ) -> X e. Fin )
701ad2antrr 492 . . . . . 6 |- ( ( ( ph /\ g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) /\ ( a e. ( 1 ... (♯ ` X ) ) /\ b e. ( 1 ... (♯ ` X ) ) ) ) -> X C_ ZZ )
714ad2antrr 492 . . . . . 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 492 . . . . . 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 533 . . . . . 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 535 . . . . . 6 |- ( ( ( ph /\ g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) /\ ( a e. ( 1 ... (♯ ` X ) ) /\ b e. ( 1 ... (♯ ` X ) ) ) ) -> a e. ( 1 ... (♯ ` X ) ) )
76 simprr 537 . . . . . 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 11269 . . . . 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 2632 . . . 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 5381 . . . 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 421 . . 3 |- ( ( ph /\ g Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) ) -> ( g u. { <. (♯ ` X ) , M >. } ) Isom < , < ( ( 1 ... (♯ ` X ) ) , X ) )
81 vex 2824 . . . . . . 7 |- g e. _V
8281a1i 9 . . . . . 6 |- ( ph -> g e. _V )
83 opexg 4363 . . . . . . . 8 |- ( ( (♯ ` X ) e. NN0 /\ M e. X ) -> <. (♯ ` X ) , M >. e. _V )
8438, 4, 83syl2anc 415 . . . . . . 7 |- ( ph -> <. (♯ ` X ) , M >. e. _V )
85 snexg 4316 . . . . . . 7 |- ( <. (♯ ` X ) , M >. e. _V -> { <. (♯ ` X ) , M >. } e. _V )
8684, 85syl 14 . . . . . 6 |- ( ph -> { <. (♯ ` X ) , M >. } e. _V )
87 unexg 4584 . . . . . 6 |- ( ( g e. _V /\ { <. (♯ ` X ) , M >. } e. _V ) -> ( g u. { <. (♯ ` X ) , M >. } ) e. _V )
8882, 86, 87syl2anc 415 . . . . 5 |- ( ph -> ( g u. { <. (♯ ` X ) , M >. } ) e. _V )
89 isoeq1 5997 . . . . . 6 |- ( f = ( g u. { <. (♯ ` X ) , M >. } ) -> ( f Isom < , < ( ( 1 ... (♯ ` X ) ) , X ) <-> ( g u. { <. (♯ ` X ) , M >. } ) Isom < , < ( ( 1 ... (♯ ` X ) ) , X ) ) )
9089spcegv 2913 . . . . 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 1954 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 1400   = wceq 1402   E.wex 1545   e. wcel 2209   A.wral 2528   _Vcvv 2821   \ cdif 3217   u. cun 3218   i^i cin 3219   C_ wss 3220   (/)c0 3520   {csn 3705   <.cop 3708   class class class wbr 4125   suc csuc 4505   omcom 4732   -1-1-onto->wf1o 5371   ` cfv 5372   Isom wiso 5373  (class class class)co 6075   ~~ cen 7010   Fincfn 7012   1c1 8170   + caddc 8172   < clt 8350   <_ cle 8351   - cmin 8487   NN0cn0 9542   ZZcz 9623   ...cfz 10390  ♯chash 11192
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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285
This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-isom 5381  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-frec 6652  df-1o 6677  df-oadd 6681  df-er 6797  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391  df-ihash 11193
This theorem is referenced by:  zfz1iso  11271
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