Theorem fseqdom 8849

Description: One half of fseqen 8850. (Contributed by Mario Carneiro, 18-Nov-2014.)

Assertion

Ref	Expression
fseqdom	|- ( A e. V -> ( om X. A ) ~<_ U_ n e. om ( A ^m n ) )

Distinct variable group:   A, n

Allowed substitution hint:   V( n)

Proof of Theorem fseqdom

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		omex 8540	. . 3 |- om e. _V
2		ovex 6678	. . 3 |- ( A ^m n ) e. _V
3	1, 2	iunex 7147	. 2 |- U_ n e. om ( A ^m n ) e. _V
4		xp1st 7198	. . . . . 6 |- ( x e. ( om X. A ) -> ( 1st ` x ) e. om )
5		peano2 7086	. . . . . 6 |- ( ( 1st ` x ) e. om -> suc ( 1st ` x ) e. om )
6	4, 5	syl 17	. . . . 5 |- ( x e. ( om X. A ) -> suc ( 1st ` x ) e. om )
7		xp2nd 7199	. . . . . . . 8 |- ( x e. ( om X. A ) -> ( 2nd ` x ) e. A )
8		fconst6g 6094	. . . . . . . 8 |- ( ( 2nd ` x ) e. A -> ( suc ( 1st ` x ) X. { ( 2nd ` x ) } ) : suc ( 1st ` x ) --> A )
9	7, 8	syl 17	. . . . . . 7 |- ( x e. ( om X. A ) -> ( suc ( 1st ` x ) X. { ( 2nd ` x ) } ) : suc ( 1st ` x ) --> A )
10	9	adantl 482	. . . . . 6 |- ( ( A e. V /\ x e. ( om X. A ) ) -> ( suc ( 1st ` x ) X. { ( 2nd ` x ) } ) : suc ( 1st ` x ) --> A )
11		elmapg 7870	. . . . . . 7 |- ( ( A e. V /\ suc ( 1st ` x ) e. om ) -> ( ( suc ( 1st ` x ) X. { ( 2nd ` x ) } ) e. ( A ^m suc ( 1st ` x ) ) <-> ( suc ( 1st ` x ) X. { ( 2nd ` x ) } ) : suc ( 1st ` x ) --> A ) )
12	6, 11	sylan2 491	. . . . . 6 |- ( ( A e. V /\ x e. ( om X. A ) ) -> ( ( suc ( 1st ` x ) X. { ( 2nd ` x ) } ) e. ( A ^m suc ( 1st ` x ) ) <-> ( suc ( 1st ` x ) X. { ( 2nd ` x ) } ) : suc ( 1st ` x ) --> A ) )
13	10, 12	mpbird 247	. . . . 5 |- ( ( A e. V /\ x e. ( om X. A ) ) -> ( suc ( 1st ` x ) X. { ( 2nd ` x ) } ) e. ( A ^m suc ( 1st ` x ) ) )
14		oveq2 6658	. . . . . 6 |- ( n = suc ( 1st ` x ) -> ( A ^m n ) = ( A ^m suc ( 1st ` x ) ) )
15	14	eliuni 4526	. . . . 5 |- ( ( suc ( 1st ` x ) e. om /\ ( suc ( 1st ` x ) X. { ( 2nd ` x ) } ) e. ( A ^m suc ( 1st ` x ) ) ) -> ( suc ( 1st ` x ) X. { ( 2nd ` x ) } ) e. U_ n e. om ( A ^m n ) )
16	6, 13, 15	syl2an2 875	. . . 4 |- ( ( A e. V /\ x e. ( om X. A ) ) -> ( suc ( 1st ` x ) X. { ( 2nd ` x ) } ) e. U_ n e. om ( A ^m n ) )
17	16	ex 450	. . 3 |- ( A e. V -> ( x e. ( om X. A ) -> ( suc ( 1st ` x ) X. { ( 2nd ` x ) } ) e. U_ n e. om ( A ^m n ) ) )
18		nsuceq0 5805	. . . . . . 7 |- suc ( 1st ` x ) =/= (/)
19		fvex 6201	. . . . . . . 8 |- ( 2nd ` x ) e. _V
20	19	snnz 4309	. . . . . . 7 |- { ( 2nd ` x ) } =/= (/)
21		xp11 5569	. . . . . . 7 |- ( ( suc ( 1st ` x ) =/= (/) /\ { ( 2nd ` x ) } =/= (/) ) -> ( ( suc ( 1st ` x ) X. { ( 2nd ` x ) } ) = ( suc ( 1st ` y ) X. { ( 2nd ` y ) } ) <-> ( suc ( 1st ` x ) = suc ( 1st ` y ) /\ { ( 2nd ` x ) } = { ( 2nd ` y ) } ) ) )
22	18, 20, 21	mp2an 708	. . . . . 6 |- ( ( suc ( 1st ` x ) X. { ( 2nd ` x ) } ) = ( suc ( 1st ` y ) X. { ( 2nd ` y ) } ) <-> ( suc ( 1st ` x ) = suc ( 1st ` y ) /\ { ( 2nd ` x ) } = { ( 2nd ` y ) } ) )
23		xp1st 7198	. . . . . . . 8 |- ( y e. ( om X. A ) -> ( 1st ` y ) e. om )
24		peano4 7088	. . . . . . . 8 |- ( ( ( 1st ` x ) e. om /\ ( 1st ` y ) e. om ) -> ( suc ( 1st ` x ) = suc ( 1st ` y ) <-> ( 1st ` x ) = ( 1st ` y ) ) )
25	4, 23, 24	syl2an 494	. . . . . . 7 |- ( ( x e. ( om X. A ) /\ y e. ( om X. A ) ) -> ( suc ( 1st ` x ) = suc ( 1st ` y ) <-> ( 1st ` x ) = ( 1st ` y ) ) )
26		sneqbg 4374	. . . . . . . 8 |- ( ( 2nd ` x ) e. _V -> ( { ( 2nd ` x ) } = { ( 2nd ` y ) } <-> ( 2nd ` x ) = ( 2nd ` y ) ) )
27	19, 26	mp1i 13	. . . . . . 7 |- ( ( x e. ( om X. A ) /\ y e. ( om X. A ) ) -> ( { ( 2nd ` x ) } = { ( 2nd ` y ) } <-> ( 2nd ` x ) = ( 2nd ` y ) ) )
28	25, 27	anbi12d 747	. . . . . 6 |- ( ( x e. ( om X. A ) /\ y e. ( om X. A ) ) -> ( ( suc ( 1st ` x ) = suc ( 1st ` y ) /\ { ( 2nd ` x ) } = { ( 2nd ` y ) } ) <-> ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) = ( 2nd ` y ) ) ) )
29	22, 28	syl5bb 272	. . . . 5 |- ( ( x e. ( om X. A ) /\ y e. ( om X. A ) ) -> ( ( suc ( 1st ` x ) X. { ( 2nd ` x ) } ) = ( suc ( 1st ` y ) X. { ( 2nd ` y ) } ) <-> ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) = ( 2nd ` y ) ) ) )
30		xpopth 7207	. . . . 5 |- ( ( x e. ( om X. A ) /\ y e. ( om X. A ) ) -> ( ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) = ( 2nd ` y ) ) <-> x = y ) )
31	29, 30	bitrd 268	. . . 4 |- ( ( x e. ( om X. A ) /\ y e. ( om X. A ) ) -> ( ( suc ( 1st ` x ) X. { ( 2nd ` x ) } ) = ( suc ( 1st ` y ) X. { ( 2nd ` y ) } ) <-> x = y ) )
32	31	a1i 11	. . 3 |- ( A e. V -> ( ( x e. ( om X. A ) /\ y e. ( om X. A ) ) -> ( ( suc ( 1st ` x ) X. { ( 2nd ` x ) } ) = ( suc ( 1st ` y ) X. { ( 2nd ` y ) } ) <-> x = y ) ) )
33	17, 32	dom2d 7996	. 2 |- ( A e. V -> ( U_ n e. om ( A ^m n ) e. _V -> ( om X. A ) ~<_ U_ n e. om ( A ^m n ) ) )
34	3, 33	mpi 20	1 |- ( A e. V -> ( om X. A ) ~<_ U_ n e. om ( A ^m n ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   (/)c0 3915   {csn 4177   U_ciun 4520   class class class wbr 4653   X. cxp 5112   suc csuc 5725   -->wf 5884   ` cfv 5888  (class class class)co 6650   omcom 7065   1stc1st 7166   2ndc2nd 7167   ^m cmap 7857   ~<_ cdom 7953

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-map 7859  df-dom 7957

This theorem is referenced by:  fseqen  8850