Theorem cbvfo 3891

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 groups:   x,A   y,B   x,y,F   ph,y   ps,x

Proof of Theorem cbvfo

Step	Hyp	Ref	Expression
1		fofun 3679	. . 3 |- (F:A-onto->B -> Fun F)
2		visset 1816	. . . . . . . . . . . 12 |- x e. V
3	2	breldm 3321	. . . . . . . . . . 11 |- (xFy -> x e. dom F)
4	3	a1i 8	. . . . . . . . . 10 |- (Fun F -> (xFy -> x e. dom F))
5		visset 1816	. . . . . . . . . . 11 |- y e. V
6	5	funbrfv 3756	. . . . . . . . . 10 |- (Fun F -> (xFy -> (F` x) = y))
7	4, 6	jcad 602	. . . . . . . . 9 |- (Fun F -> (xFy -> (x e. dom F /\ (F` x) = y)))
8	7	19.22dv 1292	. . . . . . . 8 |- (Fun F -> (E.x xFy -> E.x(x e. dom F /\ (F` x) = y)))
9	5	elrn 3356	. . . . . . . 8 |- (y e. ran F <-> E.x xFy)
10	8, 9	syl5ib 206	. . . . . . 7 |- (Fun F -> (y e. ran F -> E.x(x e. dom F /\ (F` x) = y)))
11		hba1 1005	. . . . . . . 8 |- (A.x(x e. dom F -> ph) -> A.xA.x(x e. dom F -> ph))
12		ax-17 973	. . . . . . . 8 |- (ps -> A.xps)
13		cbvfo.1	. . . . . . . . . . . 12 |- ((F` x) = y -> (ph <-> ps))
14	13	biimpcd 155	. . . . . . . . . . 11 |- (ph -> ((F` x) = y -> ps))
15	14	imim2i 17	. . . . . . . . . 10 |- ((x e. dom F -> ph) -> (x e. dom F -> ((F` x) = y -> ps)))
16	15	imp3a 361	. . . . . . . . 9 |- ((x e. dom F -> ph) -> ((x e. dom F /\ (F` x) = y) -> ps))
17	16	a4s 986	. . . . . . . 8 |- (A.x(x e. dom F -> ph) -> ((x e. dom F /\ (F` x) = y) -> ps))
18	11, 12, 17	19.23ad 1068	. . . . . . 7 |- (A.x(x e. dom F -> ph) -> (E.x(x e. dom F /\ (F` x) = y) -> ps))
19	10, 18	syl9 57	. . . . . 6 |- (Fun F -> (A.x(x e. dom F -> ph) -> (y e. ran F -> ps)))
20	19	19.21adv 1290	. . . . 5 |- (Fun F -> (A.x(x e. dom F -> ph) -> A.y(y e. ran F -> ps)))
21	2, 5	brelrn 3350	. . . . . . . . . . 11 |- (xFy -> y e. ran F)
22	21	a1i 8	. . . . . . . . . 10 |- (Fun F -> (xFy -> y e. ran F))
23	22, 6	jcad 602	. . . . . . . . 9 |- (Fun F -> (xFy -> (y e. ran F /\ (F` x) = y)))
24	23	19.22dv 1292	. . . . . . . 8 |- (Fun F -> (E.y xFy -> E.y(y e. ran F /\ (F` x) = y)))
25	2	eldm 3313	. . . . . . . 8 |- (x e. dom F <-> E.y xFy)
26	24, 25	syl5ib 206	. . . . . . 7 |- (Fun F -> (x e. dom F -> E.y(y e. ran F /\ (F` x) = y)))
27		hba1 1005	. . . . . . . 8 |- (A.y(y e. ran F -> ps) -> A.yA.y(y e. ran F -> ps))
28		ax-17 973	. . . . . . . 8 |- (ph -> A.yph)
29	13	biimprcd 156	. . . . . . . . . . 11 |- (ps -> ((F` x) = y -> ph))
30	29	imim2i 17	. . . . . . . . . 10 |- ((y e. ran F -> ps) -> (y e. ran F -> ((F` x) = y -> ph)))
31	30	imp3a 361	. . . . . . . . 9 |- ((y e. ran F -> ps) -> ((y e. ran F /\ (F` x) = y) -> ph))
32	31	a4s 986	. . . . . . . 8 |- (A.y(y e. ran F -> ps) -> ((y e. ran F /\ (F` x) = y) -> ph))
33	27, 28, 32	19.23ad 1068	. . . . . . 7 |- (A.y(y e. ran F -> ps) -> (E.y(y e. ran F /\ (F` x) = y) -> ph))
34	26, 33	syl9 57	. . . . . 6 |- (Fun F -> (A.y(y e. ran F -> ps) -> (x e. dom F -> ph)))
35	34	19.21adv 1290	. . . . 5 |- (Fun F -> (A.y(y e. ran F -> ps) -> A.x(x e. dom F -> ph)))
36	20, 35	impbid 518	. . . 4 |- (Fun F -> (A.x(x e. dom F -> ph) <-> A.y(y e. ran F -> ps)))
37		df-ral 1652	. . . 4 |- (A.x e. dom Fph <-> A.x(x e. dom F -> ph))
38		df-ral 1652	. . . 4 |- (A.y e. ran Fps <-> A.y(y e. ran F -> ps))
39	36, 37, 38	3bitr4g 557	. . 3 |- (Fun F -> (A.x e. dom Fph <-> A.y e. ran Fps))
40	1, 39	syl 10	. 2 |- (F:A-onto->B -> (A.x e. dom Fph <-> A.y e. ran Fps))
41		fof 3678	. . 3 |- (F:A-onto->B -> F:A-->B)
42		fdm 3637	. . 3 |- (F:A-->B -> dom F = A)
43		raleq1 1789	. . 3 |- (dom F = A -> (A.x e. dom Fph <-> A.x e. A ph))
44	41, 42, 43	3syl 20	. 2 |- (F:A-onto->B -> (A.x e. dom Fph <-> A.x e. A ph))
45		forn 3680	. . 3 |- (F:A-onto->B -> ran F = B)
46	45	raleq1d 1792	. 2 |- (F:A-onto->B -> (A.y e. ran Fps <-> A.y e. B ps))
47	40, 44, 46	3bitr3d 550	1 |- (F:A-onto->B -> (A.x e. A ph <-> A.y e. B ps))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 956   = wceq 958   e. wcel 960  E.wex 982  A.wral 1648   class class class wbr 2624  dom cdm 3176  ran crn 3177  Fun wfun 3182  -->wf 3184  -onto->wfo 3186  ` cfv 3188

This theorem is referenced by:  cbvexfo 3892  isowe 3909  f1oweALT 3912

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fo 3202  df-fv 3204

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