Theorem cbvfo 2923

Description: Change bound variable between domain and range of function.

Hypothesis

Ref	Expression
cbvfo.1	|- ((F` x) = y -> (ph <-> ps))

Assertion

Ref	Expression
cbvfo	|- (F:A-onto->B -> (A.x e. A ph <-> A.y e. B ps))

Distinct variable group(s):   x,y,A   x,B,y   x,F,y   ph,y   ps,x

Proof of Theorem cbvfo

Step	Hyp	Ref	Expression
1		fof 2788	. . 3 |- (F:A-onto->B -> F:A-->B)
2		ffun 2754	. . 3 |- (F:A-->B -> Fun F)
3		visset 1350	. . . . . . . . . . . 12 |- x e. V
4	3	breldm 2535	. . . . . . . . . . 11 |- (xFy -> x e. dom F)
5	4	a1i 7	. . . . . . . . . 10 |- (Fun F -> (xFy -> x e. dom F))
6		visset 1350	. . . . . . . . . . 11 |- y e. V
7	6	funbrfv 2852	. . . . . . . . . 10 |- (Fun F -> (xFy -> (F` x) = y))
8	5, 7	jcad 455	. . . . . . . . 9 |- (Fun F -> (xFy -> (x e. dom F /\ (F` x) = y)))
9	8	19.22dv 947	. . . . . . . 8 |- (Fun F -> (E.x xFy -> E.x(x e. dom F /\ (F` x) = y)))
10	6	elrn2 2563	. . . . . . . 8 |- (y e. ran F <-> E.x xFy)
11	9, 10	syl5ib 181	. . . . . . 7 |- (Fun F -> (y e. ran F -> E.x(x e. dom F /\ (F` x) = y)))
12		hba1 698	. . . . . . . 8 |- (A.x(x e. dom F -> ph) -> A.xA.x(x e. dom F -> ph))
13		ax-17 925	. . . . . . . 8 |- (ps -> A.xps)
14		cbvfo.1	. . . . . . . . . . . 12 |- ((F` x) = y -> (ph <-> ps))
15	14	biimpcd 137	. . . . . . . . . . 11 |- (ph -> ((F` x) = y -> ps))
16	15	syl3 18	. . . . . . . . . 10 |- ((x e. dom F -> ph) -> (x e. dom F -> ((F` x) = y -> ps)))
17	16	imp3a 279	. . . . . . . . 9 |- ((x e. dom F -> ph) -> ((x e. dom F /\ (F` x) = y) -> ps))
18	17	a4s 682	. . . . . . . 8 |- (A.x(x e. dom F -> ph) -> ((x e. dom F /\ (F` x) = y) -> ps))
19	12, 13, 18	19.23ad 748	. . . . . . 7 |- (A.x(x e. dom F -> ph) -> (E.x(x e. dom F /\ (F` x) = y) -> ps))
20	11, 19	syl9 55	. . . . . 6 |- (Fun F -> (A.x(x e. dom F -> ph) -> (y e. ran F -> ps)))
21	20	19.21adv 945	. . . . 5 |- (Fun F -> (A.x(x e. dom F -> ph) -> A.y(y e. ran F -> ps)))
22	3, 6	brelrn 2559	. . . . . . . . . . 11 |- (xFy -> y e. ran F)
23	22	a1i 7	. . . . . . . . . 10 |- (Fun F -> (xFy -> y e. ran F))
24	23, 7	jcad 455	. . . . . . . . 9 |- (Fun F -> (xFy -> (y e. ran F /\ (F` x) = y)))
25	24	19.22dv 947	. . . . . . . 8 |- (Fun F -> (E.y xFy -> E.y(y e. ran F /\ (F` x) = y)))
26	3	eldm 2527	. . . . . . . 8 |- (x e. dom F <-> E.y xFy)
27	25, 26	syl5ib 181	. . . . . . 7 |- (Fun F -> (x e. dom F -> E.y(y e. ran F /\ (F` x) = y)))
28		hba1 698	. . . . . . . 8 |- (A.y(y e. ran F -> ps) -> A.yA.y(y e. ran F -> ps))
29		ax-17 925	. . . . . . . 8 |- (ph -> A.yph)
30	14	biimprcd 138	. . . . . . . . . . 11 |- (ps -> ((F` x) = y -> ph))
31	30	syl3 18	. . . . . . . . . 10 |- ((y e. ran F -> ps) -> (y e. ran F -> ((F` x) = y -> ph)))
32	31	imp3a 279	. . . . . . . . 9 |- ((y e. ran F -> ps) -> ((y e. ran F /\ (F` x) = y) -> ph))
33	32	a4s 682	. . . . . . . 8 |- (A.y(y e. ran F -> ps) -> ((y e. ran F /\ (F` x) = y) -> ph))
34	28, 29, 33	19.23ad 748	. . . . . . 7 |- (A.y(y e. ran F -> ps) -> (E.y(y e. ran F /\ (F` x) = y) -> ph))
35	27, 34	syl9 55	. . . . . 6 |- (Fun F -> (A.y(y e. ran F -> ps) -> (x e. dom F -> ph)))
36	35	19.21adv 945	. . . . 5 |- (Fun F -> (A.y(y e. ran F -> ps) -> A.x(x e. dom F -> ph)))
37	21, 36	impbid 397	. . . 4 |- (Fun F -> (A.x(x e. dom F -> ph) <-> A.y(y e. ran F -> ps)))
38		df-ral 1205	. . . 4 |- (A.x e. dom Fph <-> A.x(x e. dom F -> ph))
39		df-ral 1205	. . . 4 |- (A.y e. ran Fps <-> A.y(y e. ran F -> ps))
40	37, 38, 39	3bitr4g 428	. . 3 |- (Fun F -> (A.x e. dom Fph <-> A.y e. ran Fps))
41	1, 2, 40	3syl 21	. 2 |- (F:A-onto->B -> (A.x e. dom Fph <-> A.y e. ran Fps))
42		fdm 2756	. . 3 |- (F:A-->B -> dom F = A)
43		raleq 1324	. . 3 |- (dom F = A -> (A.x e. dom Fph <-> A.x e. A ph))
44	1, 42, 43	3syl 21	. 2 |- (F:A-onto->B -> (A.x e. dom Fph <-> A.x e. A ph))
45		forn 2789	. . 3 |- (F:A-onto->B -> ran F = B)
46		raleq 1324	. . 3 |- (ran F = B -> (A.y e. ran Fps <-> A.y e. B ps))
47	45, 46	syl 12	. 2 |- (F:A-onto->B -> (A.y e. ran Fps <-> A.y e. B ps))
48	41, 44, 47	3bitr3d 423	1 |- (F:A-onto->B -> (A.x e. A ph <-> A.y e. B ps))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678   = wceq 1091   e. wcel 1092  A.wral 1201   class class class wbr 2054  dom cdm 2410  ran crn 2411  Fun wfun 2416  -->wf 2418  -onto->wfo 2420  ` cfv 2422

This theorem is referenced by:  cbvexfo 2924  isowe 2941  f1oweOLD 2944

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-fo 2436  df-fv 2438

metamath.org