Theorem dffv2 4734

Description: Alternate definition of function value df-fv 4014 that doesn't require dummy variables.

Assertion

Ref	Expression
dffv2	|- (F` A) = U.((F"{A}) \ U.U.(((F |` {A}) o. `'(F |` {A})) \ _I ))

Proof of Theorem dffv2

Step	Hyp	Ref	Expression
1		funfv 4731	. . . 4 |- (Fun (F |` {A}) -> ((F |` {A})` A) = U.((F |` {A})"{A}))
2		df-fun 4008	. . . . . . . . . . . . 13 |- (Fun (F |` {A}) <-> (Rel (F |` {A}) /\ ((F |` {A}) o. `'(F |` {A})) C_ _I ))
3	2	simprbi 353	. . . . . . . . . . . 12 |- (Fun (F |` {A}) -> ((F |` {A}) o. `'(F |` {A})) C_ _I )
4		ssdif0 2934	. . . . . . . . . . . 12 |- (((F |` {A}) o. `'(F |` {A})) C_ _I <-> (((F |` {A}) o. `'(F |` {A})) \ _I ) = (/))
5	3, 4	sylib 215	. . . . . . . . . . 11 |- (Fun (F |` {A}) -> (((F |` {A}) o. `'(F |` {A})) \ _I ) = (/))
6	5	unieqd 3188	. . . . . . . . . 10 |- (Fun (F |` {A}) -> U.(((F |` {A}) o. `'(F |` {A})) \ _I ) = U.(/))
7		uni0 3205	. . . . . . . . . 10 |- U.(/) = (/)
8	6, 7	syl6eq 1944	. . . . . . . . 9 |- (Fun (F |` {A}) -> U.(((F |` {A}) o. `'(F |` {A})) \ _I ) = (/))
9	8	unieqd 3188	. . . . . . . 8 |- (Fun (F |` {A}) -> U.U.(((F |` {A}) o. `'(F |` {A})) \ _I ) = U.(/))
10	9, 7	syl6eq 1944	. . . . . . 7 |- (Fun (F |` {A}) -> U.U.(((F |` {A}) o. `'(F |` {A})) \ _I ) = (/))
11	10	difeq2d 2726	. . . . . 6 |- (Fun (F |` {A}) -> ((F"{A}) \ U.U.(((F |` {A}) o. `'(F |` {A})) \ _I )) = ((F"{A}) \ (/)))
12		resima 4247	. . . . . . 7 |- ((F |` {A})"{A}) = (F"{A})
13		dif0 2943	. . . . . . 7 |- ((F"{A}) \ (/)) = (F"{A})
14	12, 13	eqtr4i 1911	. . . . . 6 |- ((F |` {A})"{A}) = ((F"{A}) \ (/))
15	11, 14	syl6reqr 1947	. . . . 5 |- (Fun (F |` {A}) -> ((F |` {A})"{A}) = ((F"{A}) \ U.U.(((F |` {A}) o. `'(F |` {A})) \ _I )))
16	15	unieqd 3188	. . . 4 |- (Fun (F |` {A}) -> U.((F |` {A})"{A}) = U.((F"{A}) \ U.U.(((F |` {A}) o. `'(F |` {A})) \ _I )))
17	1, 16	eqtrd 1925	. . 3 |- (Fun (F |` {A}) -> ((F |` {A})` A) = U.((F"{A}) \ U.U.(((F |` {A}) o. `'(F |` {A})) \ _I )))
18		snidb 3068	. . . . 5 |- (A e. _V <-> A e. {A})
19		fvres 4691	. . . . 5 |- (A e. {A} -> ((F |` {A})` A) = (F` A))
20	18, 19	sylbi 216	. . . 4 |- (A e. _V -> ((F |` {A})` A) = (F` A))
21		fvprc 4678	. . . . 5 |- (-. A e. _V -> ((F |` {A})` A) = (/))
22		fvprc 4678	. . . . 5 |- (-. A e. _V -> (F` A) = (/))
23	21, 22	eqtr4d 1928	. . . 4 |- (-. A e. _V -> ((F |` {A})` A) = (F` A))
24	20, 23	pm2.61i 140	. . 3 |- ((F |` {A})` A) = (F` A)
25	17, 24	syl5eqr 1942	. 2 |- (Fun (F |` {A}) -> (F` A) = U.((F"{A}) \ U.U.(((F |` {A}) o. `'(F |` {A})) \ _I )))
26		nfunsn 4706	. . 3 |- (-. Fun (F |` {A}) -> (F` A) = (/))
27		dffun3 4432	. . . . . . . . . . . . . . 15 |- (Fun (F |` {A}) <-> (Rel (F |` {A}) /\ A.xE.yA.z(x(F |` {A})z -> z = y)))
28		relres 4242	. . . . . . . . . . . . . . 15 |- Rel (F |` {A})
29	27, 28	mpbiran 798	. . . . . . . . . . . . . 14 |- (Fun (F |` {A}) <-> A.xE.yA.z(x(F |` {A})z -> z = y))
30		iman 256	. . . . . . . . . . . . . . . . . . 19 |- ((x(F |` {A})z -> z = y) <-> -. (x(F |` {A})z /\ -. z = y))
31	30	albii 1346	. . . . . . . . . . . . . . . . . 18 |- (A.z(x(F |` {A})z -> z = y) <-> A.z -. (x(F |` {A})z /\ -. z = y))
32		alnex 1380	. . . . . . . . . . . . . . . . . 18 |- (A.z -. (x(F |` {A})z /\ -. z = y) <-> -. E.z(x(F |` {A})z /\ -. z = y))
33	31, 32	bitri 190	. . . . . . . . . . . . . . . . 17 |- (A.z(x(F |` {A})z -> z = y) <-> -. E.z(x(F |` {A})z /\ -. z = y))
34	33	exbii 1398	. . . . . . . . . . . . . . . 16 |- (E.yA.z(x(F |` {A})z -> z = y) <-> E.y -. E.z(x(F |` {A})z /\ -. z = y))
35		exnal 1385	. . . . . . . . . . . . . . . 16 |- (E.y -. E.z(x(F |` {A})z /\ -. z = y) <-> -. A.yE.z(x(F |` {A})z /\ -. z = y))
36	34, 35	bitri 190	. . . . . . . . . . . . . . 15 |- (E.yA.z(x(F |` {A})z -> z = y) <-> -. A.yE.z(x(F |` {A})z /\ -. z = y))
37	36	albii 1346	. . . . . . . . . . . . . 14 |- (A.xE.yA.z(x(F |` {A})z -> z = y) <-> A.x -. A.yE.z(x(F |` {A})z /\ -. z = y))
38		alnex 1380	. . . . . . . . . . . . . 14 |- (A.x -. A.yE.z(x(F |` {A})z /\ -. z = y) <-> -. E.xA.yE.z(x(F |` {A})z /\ -. z = y))
39	29, 37, 38	3bitrri 195	. . . . . . . . . . . . 13 |- (-. E.xA.yE.z(x(F |` {A})z /\ -. z = y) <-> Fun (F |` {A}))
40	39	con1bii 237	. . . . . . . . . . . 12 |- (-. Fun (F |` {A}) <-> E.xA.yE.z(x(F |` {A})z /\ -. z = y))
41		ax-4 1319	. . . . . . . . . . . . 13 |- (A.yE.z(x(F |` {A})z /\ -. z = y) -> E.z(x(F |` {A})z /\ -. z = y))
42	41	eximi 1387	. . . . . . . . . . . 12 |- (E.xA.yE.z(x(F |` {A})z /\ -. z = y) -> E.xE.z(x(F |` {A})z /\ -. z = y))
43	40, 42	sylbi 216	. . . . . . . . . . 11 |- (-. Fun (F |` {A}) -> E.xE.z(x(F |` {A})z /\ -. z = y))
44		eleq2 1958	. . . . . . . . . . . . . . . . . . . . 21 |- (dom ( F |` {A}) = {A} -> (x e. dom ( F |` {A}) <-> x e. {A}))
45		elsn 3058	. . . . . . . . . . . . . . . . . . . . 21 |- (x e. {A} <-> x = A)
46	44, 45	syl6bb 595	. . . . . . . . . . . . . . . . . . . 20 |- (dom ( F |` {A}) = {A} -> (x e. dom ( F |` {A}) <-> x = A))
47	46	biimpa 460	. . . . . . . . . . . . . . . . . . 19 |- ((dom ( F |` {A}) = {A} /\ x e. dom ( F |` {A})) -> x = A)
48		snssi 3129	. . . . . . . . . . . . . . . . . . . 20 |- (A e. dom ( F |` {A}) -> {A} C_ dom ( F |` {A}))
49		ssdmres 4235	. . . . . . . . . . . . . . . . . . . . . 22 |- ({A} C_ dom ( F |` {A}) <-> dom ((F |` {A}) |` {A}) = {A})
50	49	biimpi 168	. . . . . . . . . . . . . . . . . . . . 21 |- ({A} C_ dom ( F |` {A}) -> dom ((F |` {A}) |` {A}) = {A})
51		residm 4246	. . . . . . . . . . . . . . . . . . . . . 22 |- ((F |` {A}) |` {A}) = (F |` {A})
52	51	dmeqi 4158	. . . . . . . . . . . . . . . . . . . . 21 |- dom ((F |` {A}) |` {A}) = dom ( F |` {A})
53	50, 52	syl5eqr 1942	. . . . . . . . . . . . . . . . . . . 20 |- ({A} C_ dom ( F |` {A}) -> dom ( F |` {A}) = {A})
54	48, 53	syl 12	. . . . . . . . . . . . . . . . . . 19 |- (A e. dom ( F |` {A}) -> dom ( F |` {A}) = {A})
55		visset 2295	. . . . . . . . . . . . . . . . . . . 20 |- x e. _V
56	55	breldm 4161	. . . . . . . . . . . . . . . . . . 19 |- (x(F |` {A})z -> x e. dom ( F |` {A}))
57	47, 54, 56	syl2an 503	. . . . . . . . . . . . . . . . . 18 |- ((A e. dom ( F |` {A}) /\ x(F |` {A})z) -> x = A)
58	57	breq1d 3348	. . . . . . . . . . . . . . . . 17 |- ((A e. dom ( F |` {A}) /\ x(F |` {A})z) -> (x(F |` {A})z <-> A(F |` {A})z))
59	58	biimpd 170	. . . . . . . . . . . . . . . 16 |- ((A e. dom ( F |` {A}) /\ x(F |` {A})z) -> (x(F |` {A})z -> A(F |` {A})z))
60	59	ex 402	. . . . . . . . . . . . . . 15 |- (A e. dom ( F |` {A}) -> (x(F |` {A})z -> (x(F |` {A})z -> A(F |` {A})z)))
61	60	pm2.43d 79	. . . . . . . . . . . . . 14 |- (A e. dom ( F |` {A}) -> (x(F |` {A})z -> A(F |` {A})z))
62	61	anim1d 619	. . . . . . . . . . . . 13 |- (A e. dom ( F |` {A}) -> ((x(F |` {A})z /\ -. z = y) -> (A(F |` {A})z /\ -. z = y)))
63	62	eximdv 1669	. . . . . . . . . . . 12 |- (A e. dom ( F |` {A}) -> (E.z(x(F |` {A})z /\ -. z = y) -> E.z(A(F |` {A})z /\ -. z = y)))
64	63	19.23adv 1584	. . . . . . . . . . 11 |- (A e. dom ( F |` {A}) -> (E.xE.z(x(F |` {A})z /\ -. z = y) -> E.z(A(F |` {A})z /\ -. z = y)))
65	43, 64	mpan9 521	. . . . . . . . . 10 |- ((-. Fun (F |` {A}) /\ A e. dom ( F |` {A})) -> E.z(A(F |` {A})z /\ -. z = y))
66		brcnvg 4142	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((y e. _V /\ A e. _V) -> (y`'(F |` {A})A <-> A(F |` {A})y))
67		visset 2295	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- y e. _V
68	28	brrelexi 4029	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (A(F |` {A})z -> A e. _V)
69	66, 67, 68	sylancr 526	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (A(F |` {A})z -> (y`'(F |` {A})A <-> A(F |` {A})y))
70	69	biimpar 461	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((A(F |` {A})z /\ A(F |` {A})y) -> y`'(F |` {A})A)
71	68	adantl 424	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((y`'(F |` {A})A /\ A(F |` {A})z) -> A e. _V)
72		breq2 3342	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (x = A -> (y`'(F |` {A})x <-> y`'(F |` {A})A))
73		breq1 3341	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (x = A -> (x(F |` {A})z <-> A(F |` {A})z))
74	72, 73	anbi12d 690	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (x = A -> ((y`'(F |` {A})x /\ x(F |` {A})z) <-> (y`'(F |` {A})A /\ A(F |` {A})z)))
75	74	rcla4ev 2381	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((A e. _V /\ (y`'(F |` {A})A /\ A(F |` {A})z)) -> E.x e. _V (y`'(F |` {A})x /\ x(F |` {A})z))
76	71, 75	mpancom 769	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((y`'(F |` {A})A /\ A(F |` {A})z) -> E.x e. _V (y`'(F |` {A})x /\ x(F |` {A})z))
77	76	ancoms 484	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((A(F |` {A})z /\ y`'(F |` {A})A) -> E.x e. _V (y`'(F |` {A})x /\ x(F |` {A})z))
78	70, 77	syldan 516	. . . . . . . . . . . . . . . . . . . . . 22 |- ((A(F |` {A})z /\ A(F |` {A})y) -> E.x e. _V (y`'(F |` {A})x /\ x(F |` {A})z))
79	78	anim1i 361	. . . . . . . . . . . . . . . . . . . . 21 |- (((A(F |` {A})z /\ A(F |` {A})y) /\ -. z = y) -> (E.x e. _V (y`'(F |` {A})x /\ x(F |` {A})z) /\ -. z = y))
80	79	an1rs 547	. . . . . . . . . . . . . . . . . . . 20 |- (((A(F |` {A})z /\ -. z = y) /\ A(F |` {A})y) -> (E.x e. _V (y`'(F |` {A})x /\ x(F |` {A})z) /\ -. z = y))
81		eldif 2609	. . . . . . . . . . . . . . . . . . . . 21 |- (<.y, z>. e. (((F |` {A}) o. `'(F |` {A})) \ _I ) <-> (<.y, z>. e. ((F |` {A}) o. `'(F |` {A})) /\ -. <.y, z>. e. _I ))
82		rexv 2306	. . . . . . . . . . . . . . . . . . . . . . 23 |- (E.x e. _V (y`'(F |` {A})x /\ x(F |` {A})z) <-> E.x(y`'(F |` {A})x /\ x(F |` {A})z))
83		visset 2295	. . . . . . . . . . . . . . . . . . . . . . . 24 |- z e. _V
84	67, 83	brco 4132	. . . . . . . . . . . . . . . . . . . . . . 23 |- (y((F |` {A}) o. `'(F |` {A}))z <-> E.x(y`'(F |` {A})x /\ x(F |` {A})z))
85		df-br 3339	. . . . . . . . . . . . . . . . . . . . . . 23 |- (y((F |` {A}) o. `'(F |` {A}))z <-> <.y, z>. e. ((F |` {A}) o. `'(F |` {A})))
86	82, 84, 85	3bitr2ri 197	. . . . . . . . . . . . . . . . . . . . . 22 |- (<.y, z>. e. ((F |` {A}) o. `'(F |` {A})) <-> E.x e. _V (y`'(F |` {A})x /\ x(F |` {A})z))
87	83	ideq 4116	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (y _I z <-> y = z)
88		df-br 3339	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (y _I z <-> <.y, z>. e. _I )
89		equcom 1488	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (y = z <-> z = y)
90	87, 88, 89	3bitr3i 198	. . . . . . . . . . . . . . . . . . . . . . 23 |- (<.y, z>. e. _I <-> z = y)
91	90	notbii 204	. . . . . . . . . . . . . . . . . . . . . 22 |- (-. <.y, z>. e. _I <-> -. z = y)
92	86, 91	anbi12i 540	. . . . . . . . . . . . . . . . . . . . 21 |- ((<.y, z>. e. ((F |` {A}) o. `'(F |` {A})) /\ -. <.y, z>. e. _I ) <-> (E.x e. _V (y`'(F |` {A})x /\ x(F |` {A})z) /\ -. z = y))
93	81, 92	bitr2i 191	. . . . . . . . . . . . . . . . . . . 20 |- ((E.x e. _V (y`'(F |` {A})x /\ x(F |` {A})z) /\ -. z = y) <-> <.y, z>. e. (((F |` {A}) o. `'(F |` {A})) \ _I ))
94	80, 93	sylib 215	. . . . . . . . . . . . . . . . . . 19 |- (((A(F |` {A})z /\ -. z = y) /\ A(F |` {A})y) -> <.y, z>. e. (((F |` {A}) o. `'(F |` {A})) \ _I ))
95		snssi 3129	. . . . . . . . . . . . . . . . . . 19 |- (<.y, z>. e. (((F |` {A}) o. `'(F |` {A})) \ _I ) -> {<.y, z>.} C_ (((F |` {A}) o. `'(F |` {A})) \ _I ))
96		uniss 3199	. . . . . . . . . . . . . . . . . . 19 |- ({<.y, z>.} C_ (((F |` {A}) o. `'(F |` {A})) \ _I ) -> U.{<.y, z>.} C_ U.(((F |` {A}) o. `'(F |` {A})) \ _I ))
97	94, 95, 96	3syl 24	. . . . . . . . . . . . . . . . . 18 |- (((A(F |` {A})z /\ -. z = y) /\ A(F |` {A})y) -> U.{<.y, z>.} C_ U.(((F |` {A}) o. `'(F |` {A})) \ _I ))
98		opex 3527	. . . . . . . . . . . . . . . . . . 19 |- <.y, z>. e. _V
99	98	unisn 3193	. . . . . . . . . . . . . . . . . 18 |- U.{<.y, z>.} = <.y, z>.
100	97, 99	syl5ssr 2662	. . . . . . . . . . . . . . . . 17 |- (((A(F |` {A})z /\ -. z = y) /\ A(F |` {A})y) -> <.y, z>. C_ U.(((F |` {A}) o. `'(F |` {A})) \ _I ))
101		uniss 3199	. . . . . . . . . . . . . . . . 17 |- (<.y, z>. C_ U.(((F |` {A}) o. `'(F |` {A})) \ _I ) -> U.<.y, z>. C_ U.U.(((F |` {A}) o. `'(F |` {A})) \ _I ))
102	100, 101	syl 12	. . . . . . . . . . . . . . . 16 |- (((A(F |` {A})z /\ -. z = y) /\ A(F |` {A})y) -> U.<.y, z>. C_ U.U.(((F |` {A}) o. `'(F |` {A})) \ _I ))
103		uniop 3555	. . . . . . . . . . . . . . . 16 |- U.<.y, z>. = {y, z}
104	102, 103	syl5ssr 2662	. . . . . . . . . . . . . . 15 |- (((A(F |` {A})z /\ -. z = y) /\ A(F |` {A})y) -> {y, z} C_ U.U.(((F |` {A}) o. `'(F |` {A})) \ _I ))
105	67, 83	prss 3138	. . . . . . . . . . . . . . 15 |- ((y e. U.U.(((F |` {A}) o. `'(F |` {A})) \ _I ) /\ z e. U.U.(((F |` {A}) o. `'(F |` {A})) \ _I )) <-> {y, z} C_ U.U.(((F |` {A}) o. `'(F |` {A})) \ _I ))
106	104, 105	sylibr 217	. . . . . . . . . . . . . 14 |- (((A(F |` {A})z /\ -. z = y) /\ A(F |` {A})y) -> (y e. U.U.(((F |` {A}) o. `'(F |` {A})) \ _I ) /\ z e. U.U.(((F |` {A}) o. `'(F |` {A})) \ _I )))
107	106	simplld 348	. . . . . . . . . . . . 13 |- (((A(F |` {A})z /\ -. z = y) /\ A(F |` {A})y) -> y e. U.U.(((F |` {A}) o. `'(F |` {A})) \ _I ))
108	107	ex 402	. . . . . . . . . . . 12 |- ((A(F |` {A})z /\ -. z = y) -> (A(F |` {A})y -> y e. U.U.(((F |` {A}) o. `'(F |` {A})) \ _I )))
109	12	eleq2i 1961	. . . . . . . . . . . . 13 |- (y e. ((F |` {A})"{A}) <-> y e. (F"{A}))
110		elimasni 4292	. . . . . . . . . . . . 13 |- (y e. ((F |` {A})"{A}) -> A(F |` {A})y)
111	109, 110	sylbir 218	. . . . . . . . . . . 12 |- (y e. (F"{A}) -> A(F |` {A})y)
112	108, 111	syl5 20	. . . . . . . . . . 11 |- ((A(F |` {A})z /\ -. z = y) -> (y e. (F"{A}) -> y e. U.U.(((F |` {A}) o. `'(F |` {A})) \ _I )))
113	112	19.23aiv 1674	. . . . . . . . . 10 |- (E.z(A(F |` {A})z /\ -. z = y) -> (y e. (F"{A}) -> y e. U.U.(((F |` {A}) o. `'(F |` {A})) \ _I )))
114	65, 113	syl 12	. . . . . . . . 9 |- ((-. Fun (F |` {A}) /\ A e. dom ( F |` {A})) -> (y e. (F"{A}) -> y e. U.U.(((F |` {A}) o. `'(F |` {A})) \ _I )))
115	114	ssrdv 2622	. . . . . . . 8 |- ((-. Fun (F |` {A}) /\ A e. dom ( F |` {A})) -> (F"{A}) C_ U.U.(((F |` {A}) o. `'(F |` {A})) \ _I ))
116		ssdif0 2934	. . . . . . . 8 |- ((F"{A}) C_ U.U.(((F |` {A}) o. `'(F |` {A})) \ _I ) <-> ((F"{A}) \ U.U.(((F |` {A}) o. `'(F |` {A})) \ _I )) = (/))
117	115, 116	sylib 215	. . . . . . 7 |- ((-. Fun (F |` {A}) /\ A e. dom ( F |` {A})) -> ((F"{A}) \ U.U.(((F |` {A}) o. `'(F |` {A})) \ _I )) = (/))
118	117	ex 402	. . . . . 6 |- (-. Fun (F |` {A}) -> (A e. dom ( F |` {A}) -> ((F"{A}) \ U.U.(((F |` {A}) o. `'(F |` {A})) \ _I )) = (/)))
119		ndmima 4300	. . . . . . . . 9 |- (-. A e. dom ( F |` {A}) -> ((F |` {A})"{A}) = (/))
120	119, 12	syl5eqr 1942	. . . . . . . 8 |- (-. A e. dom ( F |` {A}) -> (F"{A}) = (/))
121	120	difeq1d 2725	. . . . . . 7 |- (-. A e. dom ( F |` {A}) -> ((F"{A}) \ U.U.(((F |` {A}) o. `'(F |` {A})) \ _I )) = ((/) \ U.U.(((F |` {A}) o. `'(F |` {A})) \ _I )))
122		0dif 2944	. . . . . . 7 |- ((/) \ U.U.(((F |` {A}) o. `'(F |` {A})) \ _I )) = (/)
123	121, 122	syl6eq 1944	. . . . . 6 |- (-. A e. dom ( F |` {A}) -> ((F"{A}) \ U.U.(((F |` {A}) o. `'(F |` {A})) \ _I )) = (/))
124	118, 123	pm2.61d1 142	. . . . 5 |- (-. Fun (F |` {A}) -> ((F"{A}) \ U.U.(((F |` {A}) o. `'(F |` {A})) \ _I )) = (/))
125	124	unieqd 3188	. . . 4 |- (-. Fun (F |` {A}) -> U.((F"{A}) \ U.U.(((F |` {A}) o. `'(F |` {A})) \ _I )) = U.(/))
126	125, 7	syl6eq 1944	. . 3 |- (-. Fun (F |` {A}) -> U.((F"{A}) \ U.U.(((F |` {A}) o. `'(F |` {A})) \ _I )) = (/))
127	26, 126	eqtr4d 1928	. 2 |- (-. Fun (F |` {A}) -> (F` A) = U.((F"{A}) \ U.U.(((F |` {A}) o. `'(F |` {A})) \ _I )))
128	25, 127	pm2.61i 140	1 |- (F` A) = U.((F"{A}) \ U.U.(((F |` {A}) o. `'(F |` {A})) \ _I ))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  E.wrex 2106  _Vcvv 2292   \ cdif 2590   C_ wss 2593  (/)c0 2875  {csn 3044  {cpr 3045  <.cop 3046  U.cuni 3177   class class class wbr 3338   _I cid 3582  `'ccnv 3985  dom cdm 3986   |` cres 3988  "cima 3989   o. ccom 3990  Rel wrel 3991  Fun wfun 3992  ` cfv 3998

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014

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