Theorem frege129d 38055

Description: If F is a function and (for distinct A and B) either A follows B or B follows A in the transitive closure of F, the successor of A is either B or it follows B or it comes before B in the transitive closure of F. Similar to Proposition 129 of [Frege1879] p. 83. Comparw with frege129 38286. (Contributed by RP, 16-Jul-2020.)

Hypotheses

Ref	Expression
frege129d.f	|- ( ph -> F e. _V )
frege129d.a	|- ( ph -> A e. dom F )
frege129d.c	|- ( ph -> C = ( F ` A ) )
frege129d.or	|- ( ph -> ( A ( t+ ` F ) B \/ A = B \/ B ( t+ ` F ) A ) )
frege129d.fun	|- ( ph -> Fun F )

Assertion

Ref	Expression
frege129d	|- ( ph -> ( B ( t+ ` F ) C \/ B = C \/ C ( t+ ` F ) B ) )

Proof of Theorem frege129d

Step	Hyp	Ref	Expression
1		frege129d.or	. 2 |- ( ph -> ( A ( t+ ` F ) B \/ A = B \/ B ( t+ ` F ) A ) )
2		frege129d.f	. . . . . . . 8 |- ( ph -> F e. _V )
3	2	adantr 481	. . . . . . 7 |- ( ( ph /\ A ( t+ ` F ) B ) -> F e. _V )
4		frege129d.a	. . . . . . . 8 |- ( ph -> A e. dom F )
5	4	adantr 481	. . . . . . 7 |- ( ( ph /\ A ( t+ ` F ) B ) -> A e. dom F )
6		frege129d.c	. . . . . . . 8 |- ( ph -> C = ( F ` A ) )
7	6	adantr 481	. . . . . . 7 |- ( ( ph /\ A ( t+ ` F ) B ) -> C = ( F ` A ) )
8		simpr 477	. . . . . . 7 |- ( ( ph /\ A ( t+ ` F ) B ) -> A ( t+ ` F ) B )
9		frege129d.fun	. . . . . . . 8 |- ( ph -> Fun F )
10	9	adantr 481	. . . . . . 7 |- ( ( ph /\ A ( t+ ` F ) B ) -> Fun F )
11	3, 5, 7, 8, 10	frege126d 38054	. . . . . 6 |- ( ( ph /\ A ( t+ ` F ) B ) -> ( C ( t+ ` F ) B \/ C = B \/ B ( t+ ` F ) C ) )
12		biid 251	. . . . . . 7 |- ( C ( t+ ` F ) B <-> C ( t+ ` F ) B )
13		eqcom 2629	. . . . . . 7 |- ( C = B <-> B = C )
14		biid 251	. . . . . . 7 |- ( B ( t+ ` F ) C <-> B ( t+ ` F ) C )
15	12, 13, 14	3orbi123i 1252	. . . . . 6 |- ( ( C ( t+ ` F ) B \/ C = B \/ B ( t+ ` F ) C ) <-> ( C ( t+ ` F ) B \/ B = C \/ B ( t+ ` F ) C ) )
16	11, 15	sylib 208	. . . . 5 |- ( ( ph /\ A ( t+ ` F ) B ) -> ( C ( t+ ` F ) B \/ B = C \/ B ( t+ ` F ) C ) )
17		3orcomb 1048	. . . . . 6 |- ( ( C ( t+ ` F ) B \/ B = C \/ B ( t+ ` F ) C ) <-> ( C ( t+ ` F ) B \/ B ( t+ ` F ) C \/ B = C ) )
18		3orrot 1044	. . . . . 6 |- ( ( C ( t+ ` F ) B \/ B ( t+ ` F ) C \/ B = C ) <-> ( B ( t+ ` F ) C \/ B = C \/ C ( t+ ` F ) B ) )
19	17, 18	sylbb 209	. . . . 5 |- ( ( C ( t+ ` F ) B \/ B = C \/ B ( t+ ` F ) C ) -> ( B ( t+ ` F ) C \/ B = C \/ C ( t+ ` F ) B ) )
20	16, 19	syl 17	. . . 4 |- ( ( ph /\ A ( t+ ` F ) B ) -> ( B ( t+ ` F ) C \/ B = C \/ C ( t+ ` F ) B ) )
21	20	ex 450	. . 3 |- ( ph -> ( A ( t+ ` F ) B -> ( B ( t+ ` F ) C \/ B = C \/ C ( t+ ` F ) B ) ) )
22		simpr 477	. . . . . 6 |- ( ( ph /\ A = B ) -> A = B )
23	6	eqcomd 2628	. . . . . . . . 9 |- ( ph -> ( F ` A ) = C )
24		funbrfvb 6238	. . . . . . . . . . 11 |- ( ( Fun F /\ A e. dom F ) -> ( ( F ` A ) = C <-> A F C ) )
25	24	biimpd 219	. . . . . . . . . 10 |- ( ( Fun F /\ A e. dom F ) -> ( ( F ` A ) = C -> A F C ) )
26	9, 4, 25	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( ( F ` A ) = C -> A F C ) )
27	23, 26	mpd 15	. . . . . . . 8 |- ( ph -> A F C )
28	2, 27	frege91d 38043	. . . . . . 7 |- ( ph -> A ( t+ ` F ) C )
29	28	adantr 481	. . . . . 6 |- ( ( ph /\ A = B ) -> A ( t+ ` F ) C )
30	22, 29	eqbrtrrd 4677	. . . . 5 |- ( ( ph /\ A = B ) -> B ( t+ ` F ) C )
31	30	ex 450	. . . 4 |- ( ph -> ( A = B -> B ( t+ ` F ) C ) )
32		3mix1 1230	. . . 4 |- ( B ( t+ ` F ) C -> ( B ( t+ ` F ) C \/ B = C \/ C ( t+ ` F ) B ) )
33	31, 32	syl6 35	. . 3 |- ( ph -> ( A = B -> ( B ( t+ ` F ) C \/ B = C \/ C ( t+ ` F ) B ) ) )
34	2	adantr 481	. . . . . 6 |- ( ( ph /\ B ( t+ ` F ) A ) -> F e. _V )
35		funrel 5905	. . . . . . . . 9 |- ( Fun F -> Rel F )
36	9, 35	syl 17	. . . . . . . 8 |- ( ph -> Rel F )
37		reltrclfv 13758	. . . . . . . 8 |- ( ( F e. _V /\ Rel F ) -> Rel ( t+ ` F ) )
38	2, 36, 37	syl2anc 693	. . . . . . 7 |- ( ph -> Rel ( t+ ` F ) )
39		brrelex 5156	. . . . . . 7 |- ( ( Rel ( t+ ` F ) /\ B ( t+ ` F ) A ) -> B e. _V )
40	38, 39	sylan 488	. . . . . 6 |- ( ( ph /\ B ( t+ ` F ) A ) -> B e. _V )
41		fvex 6201	. . . . . . . 8 |- ( F ` A ) e. _V
42	6, 41	syl6eqel 2709	. . . . . . 7 |- ( ph -> C e. _V )
43	42	adantr 481	. . . . . 6 |- ( ( ph /\ B ( t+ ` F ) A ) -> C e. _V )
44		elex 3212	. . . . . . . 8 |- ( A e. dom F -> A e. _V )
45	4, 44	syl 17	. . . . . . 7 |- ( ph -> A e. _V )
46	45	adantr 481	. . . . . 6 |- ( ( ph /\ B ( t+ ` F ) A ) -> A e. _V )
47		simpr 477	. . . . . 6 |- ( ( ph /\ B ( t+ ` F ) A ) -> B ( t+ ` F ) A )
48	27	adantr 481	. . . . . 6 |- ( ( ph /\ B ( t+ ` F ) A ) -> A F C )
49	34, 40, 43, 46, 47, 48	frege96d 38041	. . . . 5 |- ( ( ph /\ B ( t+ ` F ) A ) -> B ( t+ ` F ) C )
50	49	ex 450	. . . 4 |- ( ph -> ( B ( t+ ` F ) A -> B ( t+ ` F ) C ) )
51	50, 32	syl6 35	. . 3 |- ( ph -> ( B ( t+ ` F ) A -> ( B ( t+ ` F ) C \/ B = C \/ C ( t+ ` F ) B ) ) )
52	21, 33, 51	3jaod 1392	. 2 |- ( ph -> ( ( A ( t+ ` F ) B \/ A = B \/ B ( t+ ` F ) A ) -> ( B ( t+ ` F ) C \/ B = C \/ C ( t+ ` F ) B ) ) )
53	1, 52	mpd 15	1 |- ( ph -> ( B ( t+ ` F ) C \/ B = C \/ C ( t+ ` F ) B ) )

Syntax hints:   -> wi 4   /\ wa 384   \/ w3o 1036   = wceq 1483   e. wcel 1990   _Vcvv 3200   class class class wbr 4653   dom cdm 5114   Rel wrel 5119   Fun wfun 5882   ` cfv 5888   t+ctcl 13724

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-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-nel 2898  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-int 4476  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-pred 5680  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802  df-trcl 13726  df-relexp 13761

This theorem is referenced by: (None)