Theorem axcc2 9259

Description: A possibly more useful version of ax-cc using sequences instead of countable sets. The Axiom of Infinity is needed to prove this, and indeed this implies the Axiom of Infinity. (Contributed by Mario Carneiro, 8-Feb-2013.)

Assertion

Ref	Expression
axcc2	|- E. g ( g Fn om /\ A. n e. om ( ( F ` n ) =/= (/) -> ( g ` n ) e. ( F ` n ) ) )

Distinct variable group:   g, F, n

Proof of Theorem axcc2

Dummy variables f m are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcv 2764	. . 3 |- F/_ n if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) )
2		nfcv 2764	. . 3 |- F/_ m if ( ( F ` n ) = (/) , { (/) } , ( F ` n ) )
3		fveq2 6191	. . . . 5 |- ( m = n -> ( F ` m ) = ( F ` n ) )
4	3	eqeq1d 2624	. . . 4 |- ( m = n -> ( ( F ` m ) = (/) <-> ( F ` n ) = (/) ) )
5	4, 3	ifbieq2d 4111	. . 3 |- ( m = n -> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) = if ( ( F ` n ) = (/) , { (/) } , ( F ` n ) ) )
6	1, 2, 5	cbvmpt 4749	. 2 |- ( m e. om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) = ( n e. om |-> if ( ( F ` n ) = (/) , { (/) } , ( F ` n ) ) )
7		nfcv 2764	. . 3 |- F/_ n ( { m } X. ( ( m e. om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) ` m ) )
8		nfcv 2764	. . . 4 |- F/_ m { n }
9		nffvmpt1 6199	. . . 4 |- F/_ m ( ( m e. om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) ` n )
10	8, 9	nfxp 5142	. . 3 |- F/_ m ( { n } X. ( ( m e. om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) ` n ) )
11		sneq 4187	. . . 4 |- ( m = n -> { m } = { n } )
12		fveq2 6191	. . . 4 |- ( m = n -> ( ( m e. om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) ` m ) = ( ( m e. om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) ` n ) )
13	11, 12	xpeq12d 5140	. . 3 |- ( m = n -> ( { m } X. ( ( m e. om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) ` m ) ) = ( { n } X. ( ( m e. om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) ` n ) ) )
14	7, 10, 13	cbvmpt 4749	. 2 |- ( m e. om |-> ( { m } X. ( ( m e. om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) ` m ) ) ) = ( n e. om |-> ( { n } X. ( ( m e. om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) ` n ) ) )
15		nfcv 2764	. . 3 |- F/_ n ( 2nd ` ( f ` ( ( m e. om |-> ( { m } X. ( ( m e. om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) ` m ) ) ) ` m ) ) )
16		nfcv 2764	. . . 4 |- F/_ m 2nd
17		nfcv 2764	. . . . 5 |- F/_ m f
18		nffvmpt1 6199	. . . . 5 |- F/_ m ( ( m e. om |-> ( { m } X. ( ( m e. om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) ` m ) ) ) ` n )
19	17, 18	nffv 6198	. . . 4 |- F/_ m ( f ` ( ( m e. om |-> ( { m } X. ( ( m e. om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) ` m ) ) ) ` n ) )
20	16, 19	nffv 6198	. . 3 |- F/_ m ( 2nd ` ( f ` ( ( m e. om |-> ( { m } X. ( ( m e. om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) ` m ) ) ) ` n ) ) )
21		fveq2 6191	. . . . 5 |- ( m = n -> ( ( m e. om |-> ( { m } X. ( ( m e. om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) ` m ) ) ) ` m ) = ( ( m e. om |-> ( { m } X. ( ( m e. om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) ` m ) ) ) ` n ) )
22	21	fveq2d 6195	. . . 4 |- ( m = n -> ( f ` ( ( m e. om |-> ( { m } X. ( ( m e. om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) ` m ) ) ) ` m ) ) = ( f ` ( ( m e. om |-> ( { m } X. ( ( m e. om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) ` m ) ) ) ` n ) ) )
23	22	fveq2d 6195	. . 3 |- ( m = n -> ( 2nd ` ( f ` ( ( m e. om |-> ( { m } X. ( ( m e. om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) ` m ) ) ) ` m ) ) ) = ( 2nd ` ( f ` ( ( m e. om |-> ( { m } X. ( ( m e. om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) ` m ) ) ) ` n ) ) ) )
24	15, 20, 23	cbvmpt 4749	. 2 |- ( m e. om |-> ( 2nd ` ( f ` ( ( m e. om |-> ( { m } X. ( ( m e. om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) ` m ) ) ) ` m ) ) ) ) = ( n e. om |-> ( 2nd ` ( f ` ( ( m e. om |-> ( { m } X. ( ( m e. om |-> if ( ( F ` m ) = (/) , { (/) } , ( F ` m ) ) ) ` m ) ) ) ` n ) ) ) )
25	6, 14, 24	axcc2lem 9258	1 |- E. g ( g Fn om /\ A. n e. om ( ( F ` n ) =/= (/) -> ( g ` n ) e. ( F ` n ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   (/)c0 3915   ifcif 4086   {csn 4177   |-> cmpt 4729   X. cxp 5112   Fn wfn 5883   ` cfv 5888   omcom 7065   2ndc2nd 7167

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  ax-cc 9257

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-om 7066  df-2nd 7169  df-er 7742  df-en 7956

This theorem is referenced by:  axcc3  9260  acncc  9262  domtriomlem  9264