Theorem frege124d 36347

Description: If F is a function, A is the successor of X, and B follows X in the transitive closure of F, then A and B are the same or B follows A in the transitive closure of F. Similar to Proposition 124 of [Frege1879] p. 80. Compare with frege124 36577. (Contributed by RP, 16-Jul-2020.)

Hypotheses

Ref	Expression
frege124d.f	|- ( ph -> F e. _V )
frege124d.x	|- ( ph -> X e. dom F )
frege124d.a	|- ( ph -> A = ( F ` X ) )
frege124d.xb	|- ( ph -> X ( t+ ` F ) B )
frege124d.fun	|- ( ph -> Fun F )

Assertion

Ref	Expression
frege124d	|- ( ph -> ( A ( t+ ` F ) B \/ A = B ) )

Proof of Theorem frege124d

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frege124d.a	. . 3 |- ( ph -> A = ( F ` X ) )
2		frege124d.fun	. . . . 5 |- ( ph -> Fun F )
3		frege124d.xb	. . . . . . 7 |- ( ph -> X ( t+ ` F ) B )
4	1	eqcomd 2456	. . . . . . . . . . 11 |- ( ph -> ( F ` X ) = A )
5		frege124d.x	. . . . . . . . . . . 12 |- ( ph -> X e. dom F )
6		funbrfvb 5905	. . . . . . . . . . . 12 |- ( ( Fun F /\ X e. dom F ) -> ( ( F ` X ) = A <-> X F A ) )
7	2, 5, 6	syl2anc 666	. . . . . . . . . . 11 |- ( ph -> ( ( F ` X ) = A <-> X F A ) )
8	4, 7	mpbid 214	. . . . . . . . . 10 |- ( ph -> X F A )
9		funeu 5605	. . . . . . . . . 10 |- ( ( Fun F /\ X F A ) -> E! a X F a )
10	2, 8, 9	syl2anc 666	. . . . . . . . 9 |- ( ph -> E! a X F a )
11		fvex 5873	. . . . . . . . . . . . 13 |- ( F ` X ) e. _V
12	1, 11	syl6eqel 2536	. . . . . . . . . . . 12 |- ( ph -> A e. _V )
13		sbcan 3309	. . . . . . . . . . . . 13 |- ( [. A / a ]. ( X F a /\ -. a ( t+ ` F ) B ) <-> ( [. A / a ]. X F a /\ [. A / a ]. -. a ( t+ ` F ) B ) )
14		sbcbr2g 4457	. . . . . . . . . . . . . . 15 |- ( A e. _V -> ( [. A / a ]. X F a <-> X F [_ A / a ]_ a ) )
15		csbvarg 3791	. . . . . . . . . . . . . . . 16 |- ( A e. _V -> [_ A / a ]_ a = A )
16	15	breq2d 4413	. . . . . . . . . . . . . . 15 |- ( A e. _V -> ( X F [_ A / a ]_ a <-> X F A ) )
17	14, 16	bitrd 257	. . . . . . . . . . . . . 14 |- ( A e. _V -> ( [. A / a ]. X F a <-> X F A ) )
18		sbcng 3307	. . . . . . . . . . . . . . 15 |- ( A e. _V -> ( [. A / a ]. -. a ( t+ ` F ) B <-> -. [. A / a ]. a ( t+ ` F ) B ) )
19		sbcbr1g 4456	. . . . . . . . . . . . . . . . 17 |- ( A e. _V -> ( [. A / a ]. a ( t+ ` F ) B <-> [_ A / a ]_ a ( t+ ` F ) B ) )
20	15	breq1d 4411	. . . . . . . . . . . . . . . . 17 |- ( A e. _V -> ( [_ A / a ]_ a ( t+ ` F ) B <-> A ( t+ ` F ) B ) )
21	19, 20	bitrd 257	. . . . . . . . . . . . . . . 16 |- ( A e. _V -> ( [. A / a ]. a ( t+ ` F ) B <-> A ( t+ ` F ) B ) )
22	21	notbid 296	. . . . . . . . . . . . . . 15 |- ( A e. _V -> ( -. [. A / a ]. a ( t+ ` F ) B <-> -. A ( t+ ` F ) B ) )
23	18, 22	bitrd 257	. . . . . . . . . . . . . 14 |- ( A e. _V -> ( [. A / a ]. -. a ( t+ ` F ) B <-> -. A ( t+ ` F ) B ) )
24	17, 23	anbi12d 716	. . . . . . . . . . . . 13 |- ( A e. _V -> ( ( [. A / a ]. X F a /\ [. A / a ]. -. a ( t+ ` F ) B ) <-> ( X F A /\ -. A ( t+ ` F ) B ) ) )
25	13, 24	syl5bb 261	. . . . . . . . . . . 12 |- ( A e. _V -> ( [. A / a ]. ( X F a /\ -. a ( t+ ` F ) B ) <-> ( X F A /\ -. A ( t+ ` F ) B ) ) )
26	12, 25	syl 17	. . . . . . . . . . 11 |- ( ph -> ( [. A / a ]. ( X F a /\ -. a ( t+ ` F ) B ) <-> ( X F A /\ -. A ( t+ ` F ) B ) ) )
27		spesbc 3348	. . . . . . . . . . 11 |- ( [. A / a ]. ( X F a /\ -. a ( t+ ` F ) B ) -> E. a ( X F a /\ -. a ( t+ ` F ) B ) )
28	26, 27	syl6bir 233	. . . . . . . . . 10 |- ( ph -> ( ( X F A /\ -. A ( t+ ` F ) B ) -> E. a ( X F a /\ -. a ( t+ ` F ) B ) ) )
29	8, 28	mpand 680	. . . . . . . . 9 |- ( ph -> ( -. A ( t+ ` F ) B -> E. a ( X F a /\ -. a ( t+ ` F ) B ) ) )
30		eupicka 2365	. . . . . . . . 9 |- ( ( E! a X F a /\ E. a ( X F a /\ -. a ( t+ ` F ) B ) ) -> A. a ( X F a -> -. a ( t+ ` F ) B ) )
31	10, 29, 30	syl6an 548	. . . . . . . 8 |- ( ph -> ( -. A ( t+ ` F ) B -> A. a ( X F a -> -. a ( t+ ` F ) B ) ) )
32		frege124d.f	. . . . . . . . . . . . 13 |- ( ph -> F e. _V )
33		funrel 5598	. . . . . . . . . . . . . 14 |- ( Fun F -> Rel F )
34	2, 33	syl 17	. . . . . . . . . . . . 13 |- ( ph -> Rel F )
35		reltrclfv 13074	. . . . . . . . . . . . 13 |- ( ( F e. _V /\ Rel F ) -> Rel ( t+ ` F ) )
36	32, 34, 35	syl2anc 666	. . . . . . . . . . . 12 |- ( ph -> Rel ( t+ ` F ) )
37		brrelex2 4873	. . . . . . . . . . . 12 |- ( ( Rel ( t+ ` F ) /\ X ( t+ ` F ) B ) -> B e. _V )
38	36, 3, 37	syl2anc 666	. . . . . . . . . . 11 |- ( ph -> B e. _V )
39		brcog 5000	. . . . . . . . . . 11 |- ( ( X e. dom F /\ B e. _V ) -> ( X ( ( t+ ` F ) o. F ) B <-> E. a ( X F a /\ a ( t+ ` F ) B ) ) )
40	5, 38, 39	syl2anc 666	. . . . . . . . . 10 |- ( ph -> ( X ( ( t+ ` F ) o. F ) B <-> E. a ( X F a /\ a ( t+ ` F ) B ) ) )
41	40	notbid 296	. . . . . . . . 9 |- ( ph -> ( -. X ( ( t+ ` F ) o. F ) B <-> -. E. a ( X F a /\ a ( t+ ` F ) B ) ) )
42		alinexa 1712	. . . . . . . . 9 |- ( A. a ( X F a -> -. a ( t+ ` F ) B ) <-> -. E. a ( X F a /\ a ( t+ ` F ) B ) )
43	41, 42	syl6rbbr 268	. . . . . . . 8 |- ( ph -> ( A. a ( X F a -> -. a ( t+ ` F ) B ) <-> -. X ( ( t+ ` F ) o. F ) B ) )
44	31, 43	sylibd 218	. . . . . . 7 |- ( ph -> ( -. A ( t+ ` F ) B -> -. X ( ( t+ ` F ) o. F ) B ) )
45		brdif 4452	. . . . . . . 8 |- ( X ( ( t+ ` F ) \ ( ( t+ ` F ) o. F ) ) B <-> ( X ( t+ ` F ) B /\ -. X ( ( t+ ` F ) o. F ) B ) )
46	45	simplbi2 630	. . . . . . 7 |- ( X ( t+ ` F ) B -> ( -. X ( ( t+ ` F ) o. F ) B -> X ( ( t+ ` F ) \ ( ( t+ ` F ) o. F ) ) B ) )
47	3, 44, 46	sylsyld 58	. . . . . 6 |- ( ph -> ( -. A ( t+ ` F ) B -> X ( ( t+ ` F ) \ ( ( t+ ` F ) o. F ) ) B ) )
48		trclfvdecomr 36314	. . . . . . . . . . 11 |- ( F e. _V -> ( t+ ` F ) = ( F u. ( ( t+ ` F ) o. F ) ) )
49	32, 48	syl 17	. . . . . . . . . 10 |- ( ph -> ( t+ ` F ) = ( F u. ( ( t+ ` F ) o. F ) ) )
50		uncom 3577	. . . . . . . . . 10 |- ( F u. ( ( t+ ` F ) o. F ) ) = ( ( ( t+ ` F ) o. F ) u. F )
51	49, 50	syl6eq 2500	. . . . . . . . 9 |- ( ph -> ( t+ ` F ) = ( ( ( t+ ` F ) o. F ) u. F ) )
52		eqimss 3483	. . . . . . . . 9 |- ( ( t+ ` F ) = ( ( ( t+ ` F ) o. F ) u. F ) -> ( t+ ` F ) C_ ( ( ( t+ ` F ) o. F ) u. F ) )
53	51, 52	syl 17	. . . . . . . 8 |- ( ph -> ( t+ ` F ) C_ ( ( ( t+ ` F ) o. F ) u. F ) )
54		ssundif 3850	. . . . . . . 8 |- ( ( t+ ` F ) C_ ( ( ( t+ ` F ) o. F ) u. F ) <-> ( ( t+ ` F ) \ ( ( t+ ` F ) o. F ) ) C_ F )
55	53, 54	sylib 200	. . . . . . 7 |- ( ph -> ( ( t+ ` F ) \ ( ( t+ ` F ) o. F ) ) C_ F )
56	55	ssbrd 4443	. . . . . 6 |- ( ph -> ( X ( ( t+ ` F ) \ ( ( t+ ` F ) o. F ) ) B -> X F B ) )
57	47, 56	syld 45	. . . . 5 |- ( ph -> ( -. A ( t+ ` F ) B -> X F B ) )
58		funbrfv 5901	. . . . 5 |- ( Fun F -> ( X F B -> ( F ` X ) = B ) )
59	2, 57, 58	sylsyld 58	. . . 4 |- ( ph -> ( -. A ( t+ ` F ) B -> ( F ` X ) = B ) )
60		eqcom 2457	. . . 4 |- ( ( F ` X ) = B <-> B = ( F ` X ) )
61	59, 60	syl6ib 230	. . 3 |- ( ph -> ( -. A ( t+ ` F ) B -> B = ( F ` X ) ) )
62		eqtr3 2471	. . 3 |- ( ( A = ( F ` X ) /\ B = ( F ` X ) ) -> A = B )
63	1, 61, 62	syl6an 548	. 2 |- ( ph -> ( -. A ( t+ ` F ) B -> A = B ) )
64	63	orrd 380	1 |- ( ph -> ( A ( t+ ` F ) B \/ A = B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   \/ wo 370   /\ wa 371   A.wal 1441   = wceq 1443   E.wex 1662   e. wcel 1886   E!weu 2298   _Vcvv 3044   [.wsbc 3266   [_csb 3362   \ cdif 3400   u. cun 3401   C_ wss 3403   class class class wbr 4401   dom cdm 4833   o. ccom 4837   Rel wrel 4838   Fun wfun 5575   ` cfv 5581   t+ctcl 13042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-nn 10607  df-2 10665  df-n0 10867  df-z 10935  df-uz 11157  df-fz 11782  df-seq 12211  df-trcl 13044  df-relexp 13077

This theorem is referenced by:  frege126d  36348