Theorem rnelfmlem 15592

Description: Lemma for rnelfm 15593. (Contributed by Jeff Hankins, 14-Nov-2009.)

Hypothesis

Ref	Expression
rnelfm.1	|- X = U.L

Assertion

Ref	Expression
rnelfmlem	|- (((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) -> {y | E.x e. L y = (`'F"x)} e. fBas)

Distinct variable groups:   x,y,A   x,F,y   x,L,y   x,X,y   x,Y,y

Proof of Theorem rnelfmlem

Step	Hyp	Ref	Expression
1		rnelfm.1	. . . . . . . . 9 |- X = U.L
2	1	filusb 10267	. . . . . . . 8 |- (L e. Fil -> X e. L)
3	2	3ad2ant2 898	. . . . . . 7 |- ((Y e. A /\ L e. Fil /\ F:Y-->X) -> X e. L)
4	3	adantr 425	. . . . . 6 |- (((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) -> X e. L)
5		fimacnv 4783	. . . . . . . . 9 |- (F:Y-->X -> (`'F"X) = Y)
6	5	eqcomd 1889	. . . . . . . 8 |- (F:Y-->X -> Y = (`'F"X))
7	6	3ad2ant3 899	. . . . . . 7 |- ((Y e. A /\ L e. Fil /\ F:Y-->X) -> Y = (`'F"X))
8	7	adantr 425	. . . . . 6 |- (((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) -> Y = (`'F"X))
9		imaeq2 4260	. . . . . . . 8 |- (x = X -> (`'F"x) = (`'F"X))
10	9	eqeq2d 1895	. . . . . . 7 |- (x = X -> (Y = (`'F"x) <-> Y = (`'F"X)))
11	10	rcla4ev 2381	. . . . . 6 |- ((X e. L /\ Y = (`'F"X)) -> E.x e. L Y = (`'F"x))
12	4, 8, 11	syl11anc 524	. . . . 5 |- (((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) -> E.x e. L Y = (`'F"x))
13		eqeq1 1890	. . . . . . . . 9 |- (y = Y -> (y = (`'F"x) <-> Y = (`'F"x)))
14	13	rexbidv 2124	. . . . . . . 8 |- (y = Y -> (E.x e. L y = (`'F"x) <-> E.x e. L Y = (`'F"x)))
15	14	elabg 2405	. . . . . . 7 |- (Y e. A -> (Y e. {y | E.x e. L y = (`'F"x)} <-> E.x e. L Y = (`'F"x)))
16	15	3ad2ant1 897	. . . . . 6 |- ((Y e. A /\ L e. Fil /\ F:Y-->X) -> (Y e. {y | E.x e. L y = (`'F"x)} <-> E.x e. L Y = (`'F"x)))
17	16	adantr 425	. . . . 5 |- (((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) -> (Y e. {y | E.x e. L y = (`'F"x)} <-> E.x e. L Y = (`'F"x)))
18	12, 17	mpbird 213	. . . 4 |- (((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) -> Y e. {y | E.x e. L y = (`'F"x)})
19		ne0i 2881	. . . 4 |- (Y e. {y | E.x e. L y = (`'F"x)} -> {y | E.x e. L y = (`'F"x)} =/= (/))
20	18, 19	syl 12	. . 3 |- (((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) -> {y | E.x e. L y = (`'F"x)} =/= (/))
21		filesn 10268	. . . . . . 7 |- (L e. Fil -> -. (/) e. L)
22	21	3ad2ant2 898	. . . . . 6 |- ((Y e. A /\ L e. Fil /\ F:Y-->X) -> -. (/) e. L)
23	22	adantr 425	. . . . 5 |- (((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) -> -. (/) e. L)
24		ffn 4562	. . . . . . . . . . . . . . . . . . 19 |- (F:Y-->X -> F Fn Y)
25		fvelrnb 4719	. . . . . . . . . . . . . . . . . . 19 |- (F Fn Y -> (y e. ran F <-> E.z e. Y (F` z) = y))
26	24, 25	syl 12	. . . . . . . . . . . . . . . . . 18 |- (F:Y-->X -> (y e. ran F <-> E.z e. Y (F` z) = y))
27	26	3ad2ant3 899	. . . . . . . . . . . . . . . . 17 |- ((Y e. A /\ L e. Fil /\ F:Y-->X) -> (y e. ran F <-> E.z e. Y (F` z) = y))
28	27	ad2antrr 440	. . . . . . . . . . . . . . . 16 |- ((((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) /\ (x e. L /\ y e. x)) -> (y e. ran F <-> E.z e. Y (F` z) = y))
29		eleq1 1957	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((F` z) = y -> ((F` z) e. x <-> y e. x))
30	29	biimparc 463	. . . . . . . . . . . . . . . . . . . . . 22 |- ((y e. x /\ (F` z) = y) -> (F` z) e. x)
31	30	ad2ant2l 444	. . . . . . . . . . . . . . . . . . . . 21 |- (((x e. L /\ y e. x) /\ (z e. Y /\ (F` z) = y)) -> (F` z) e. x)
32	31	adantll 428	. . . . . . . . . . . . . . . . . . . 20 |- (((((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) /\ (x e. L /\ y e. x)) /\ (z e. Y /\ (F` z) = y)) -> (F` z) e. x)
33		ffun 4565	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (F:Y-->X -> Fun F)
34	33	3ad2ant3 899	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((Y e. A /\ L e. Fil /\ F:Y-->X) -> Fun F)
35	34	adantr 425	. . . . . . . . . . . . . . . . . . . . . 22 |- (((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) -> Fun F)
36	35	ad2antrr 440	. . . . . . . . . . . . . . . . . . . . 21 |- (((((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) /\ (x e. L /\ y e. x)) /\ (z e. Y /\ (F` z) = y)) -> Fun F)
37		fdm 4567	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (F:Y-->X -> dom F = Y)
38	37	eleq2d 1964	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (F:Y-->X -> (z e. dom F <-> z e. Y))
39	38	biimpar 461	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((F:Y-->X /\ z e. Y) -> z e. dom F)
40	39	3ad2antl3 1040	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((Y e. A /\ L e. Fil /\ F:Y-->X) /\ z e. Y) -> z e. dom F)
41	40	adantlr 429	. . . . . . . . . . . . . . . . . . . . . 22 |- ((((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) /\ z e. Y) -> z e. dom F)
42	41	ad2ant2r 445	. . . . . . . . . . . . . . . . . . . . 21 |- (((((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) /\ (x e. L /\ y e. x)) /\ (z e. Y /\ (F` z) = y)) -> z e. dom F)
43		fvimacnv 4778	. . . . . . . . . . . . . . . . . . . . 21 |- ((Fun F /\ z e. dom F) -> ((F` z) e. x <-> z e. (`'F"x)))
44	36, 42, 43	syl11anc 524	. . . . . . . . . . . . . . . . . . . 20 |- (((((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) /\ (x e. L /\ y e. x)) /\ (z e. Y /\ (F` z) = y)) -> ((F` z) e. x <-> z e. (`'F"x)))
45	32, 44	mpbid 212	. . . . . . . . . . . . . . . . . . 19 |- (((((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) /\ (x e. L /\ y e. x)) /\ (z e. Y /\ (F` z) = y)) -> z e. (`'F"x))
46		n0i 2880	. . . . . . . . . . . . . . . . . . . 20 |- (z e. (`'F"x) -> -. (`'F"x) = (/))
47		eqcom 1886	. . . . . . . . . . . . . . . . . . . . 21 |- ((`'F"x) = (/) <-> (/) = (`'F"x))
48	47	notbii 204	. . . . . . . . . . . . . . . . . . . 20 |- (-. (`'F"x) = (/) <-> -. (/) = (`'F"x))
49	46, 48	sylib 215	. . . . . . . . . . . . . . . . . . 19 |- (z e. (`'F"x) -> -. (/) = (`'F"x))
50	45, 49	syl 12	. . . . . . . . . . . . . . . . . 18 |- (((((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) /\ (x e. L /\ y e. x)) /\ (z e. Y /\ (F` z) = y)) -> -. (/) = (`'F"x))
51	50	exp32 408	. . . . . . . . . . . . . . . . 17 |- ((((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) /\ (x e. L /\ y e. x)) -> (z e. Y -> ((F` z) = y -> -. (/) = (`'F"x))))
52	51	r19.23adv 2215	. . . . . . . . . . . . . . . 16 |- ((((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) /\ (x e. L /\ y e. x)) -> (E.z e. Y (F` z) = y -> -. (/) = (`'F"x)))
53	28, 52	sylbid 220	. . . . . . . . . . . . . . 15 |- ((((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) /\ (x e. L /\ y e. x)) -> (y e. ran F -> -. (/) = (`'F"x)))
54	53	con2d 107	. . . . . . . . . . . . . 14 |- ((((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) /\ (x e. L /\ y e. x)) -> ((/) = (`'F"x) -> -. y e. ran F))
55	54	expr 418	. . . . . . . . . . . . 13 |- ((((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) /\ x e. L) -> (y e. x -> ((/) = (`'F"x) -> -. y e. ran F)))
56	55	com23 36	. . . . . . . . . . . 12 |- ((((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) /\ x e. L) -> ((/) = (`'F"x) -> (y e. x -> -. y e. ran F)))
57	56	impr 422	. . . . . . . . . . 11 |- ((((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) /\ (x e. L /\ (/) = (`'F"x))) -> (y e. x -> -. y e. ran F))
58	57	19.21aiv 1664	. . . . . . . . . 10 |- ((((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) /\ (x e. L /\ (/) = (`'F"x))) -> A.y(y e. x -> -. y e. ran F))
59		imnan 261	. . . . . . . . . . . . 13 |- ((y e. x -> -. y e. ran F) <-> -. (y e. x /\ y e. ran F))
60		elin 2786	. . . . . . . . . . . . . 14 |- (y e. (x i^i ran F) <-> (y e. x /\ y e. ran F))
61	60	notbii 204	. . . . . . . . . . . . 13 |- (-. y e. (x i^i ran F) <-> -. (y e. x /\ y e. ran F))
62	59, 61	bitr4i 193	. . . . . . . . . . . 12 |- ((y e. x -> -. y e. ran F) <-> -. y e. (x i^i ran F))
63	62	albii 1346	. . . . . . . . . . 11 |- (A.y(y e. x -> -. y e. ran F) <-> A.y -. y e. (x i^i ran F))
64		eq0 2889	. . . . . . . . . . 11 |- ((x i^i ran F) = (/) <-> A.y -. y e. (x i^i ran F))
65		eqcom 1886	. . . . . . . . . . 11 |- ((x i^i ran F) = (/) <-> (/) = (x i^i ran F))
66	63, 64, 65	3bitr2i 196	. . . . . . . . . 10 |- (A.y(y e. x -> -. y e. ran F) <-> (/) = (x i^i ran F))
67	58, 66	sylib 215	. . . . . . . . 9 |- ((((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) /\ (x e. L /\ (/) = (`'F"x))) -> (/) = (x i^i ran F))
68		simpll2 916	. . . . . . . . . 10 |- ((((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) /\ (x e. L /\ (/) = (`'F"x))) -> L e. Fil)
69		simprl 450	. . . . . . . . . 10 |- ((((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) /\ (x e. L /\ (/) = (`'F"x))) -> x e. L)
70		simplr 449	. . . . . . . . . 10 |- ((((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) /\ (x e. L /\ (/) = (`'F"x))) -> ran F e. L)
71		filint 10269	. . . . . . . . . 10 |- ((L e. Fil /\ x e. L /\ ran F e. L) -> (x i^i ran F) e. L)
72	68, 69, 70, 71	syl111anc 1100	. . . . . . . . 9 |- ((((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) /\ (x e. L /\ (/) = (`'F"x))) -> (x i^i ran F) e. L)
73	67, 72	eqeltrd 1971	. . . . . . . 8 |- ((((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) /\ (x e. L /\ (/) = (`'F"x))) -> (/) e. L)
74	73	exp32 408	. . . . . . 7 |- (((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) -> (x e. L -> ((/) = (`'F"x) -> (/) e. L)))
75	74	r19.23adv 2215	. . . . . 6 |- (((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) -> (E.x e. L (/) = (`'F"x) -> (/) e. L))
76		0ex 3446	. . . . . . 7 |- (/) e. _V
77		eqeq1 1890	. . . . . . . 8 |- (y = (/) -> (y = (`'F"x) <-> (/) = (`'F"x)))
78	77	rexbidv 2124	. . . . . . 7 |- (y = (/) -> (E.x e. L y = (`'F"x) <-> E.x e. L (/) = (`'F"x)))
79	76, 78	elab 2403	. . . . . 6 |- ((/) e. {y | E.x e. L y = (`'F"x)} <-> E.x e. L (/) = (`'F"x))
80	75, 79	syl5ib 223	. . . . 5 |- (((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) -> ((/) e. {y | E.x e. L y = (`'F"x)} -> (/) e. L))
81	23, 80	mtod 123	. . . 4 |- (((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) -> -. (/) e. {y | E.x e. L y = (`'F"x)})
82		df-nel 2020	. . . 4 |- ((/) e/ {y | E.x e. L y = (`'F"x)} <-> -. (/) e. {y | E.x e. L y = (`'F"x)})
83	81, 82	sylibr 217	. . 3 |- (((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) -> (/) e/ {y | E.x e. L y = (`'F"x)})
84		filint 10269	. . . . . . . . . . . . . 14 |- ((L e. Fil /\ u e. L /\ v e. L) -> (u i^i v) e. L)
85	84	3expb 1068	. . . . . . . . . . . . 13 |- ((L e. Fil /\ (u e. L /\ v e. L)) -> (u i^i v) e. L)
86	85	adantlr 429	. . . . . . . . . . . 12 |- (((L e. Fil /\ F:Y-->X) /\ (u e. L /\ v e. L)) -> (u i^i v) e. L)
87		eqidd 1885	. . . . . . . . . . . 12 |- (((L e. Fil /\ F:Y-->X) /\ (u e. L /\ v e. L)) -> (`'F"(u i^i v)) = (`'F"(u i^i v)))
88		imaeq2 4260	. . . . . . . . . . . . . 14 |- (x = (u i^i v) -> (`'F"x) = (`'F"(u i^i v)))
89	88	eqeq2d 1895	. . . . . . . . . . . . 13 |- (x = (u i^i v) -> ((`'F"(u i^i v)) = (`'F"x) <-> (`'F"(u i^i v)) = (`'F"(u i^i v))))
90	89	rcla4ev 2381	. . . . . . . . . . . 12 |- (((u i^i v) e. L /\ (`'F"(u i^i v)) = (`'F"(u i^i v))) -> E.x e. L (`'F"(u i^i v)) = (`'F"x))
91	86, 87, 90	syl11anc 524	. . . . . . . . . . 11 |- (((L e. Fil /\ F:Y-->X) /\ (u e. L /\ v e. L)) -> E.x e. L (`'F"(u i^i v)) = (`'F"x))
92	91	3adantl1 1032	. . . . . . . . . 10 |- (((Y e. A /\ L e. Fil /\ F:Y-->X) /\ (u e. L /\ v e. L)) -> E.x e. L (`'F"(u i^i v)) = (`'F"x))
93	92	ad2ant2r 445	. . . . . . . . 9 |- ((((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) /\ ((u e. L /\ v e. L) /\ (r = (`'F"u) /\ s = (`'F"v)))) -> E.x e. L (`'F"(u i^i v)) = (`'F"x))
94		cnvimass 4286	. . . . . . . . . . . . . . 15 |- (`'F"(u i^i v)) C_ dom F
95	94	a1i 8	. . . . . . . . . . . . . 14 |- (F:Y-->X -> (`'F"(u i^i v)) C_ dom F)
96	95, 37	sseqtrd 2653	. . . . . . . . . . . . 13 |- (F:Y-->X -> (`'F"(u i^i v)) C_ Y)
97	96	3ad2ant3 899	. . . . . . . . . . . 12 |- ((Y e. A /\ L e. Fil /\ F:Y-->X) -> (`'F"(u i^i v)) C_ Y)
98	97	ad2antrr 440	. . . . . . . . . . 11 |- ((((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) /\ ((u e. L /\ v e. L) /\ (r = (`'F"u) /\ s = (`'F"v)))) -> (`'F"(u i^i v)) C_ Y)
99		simpll1 915	. . . . . . . . . . 11 |- ((((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) /\ ((u e. L /\ v e. L) /\ (r = (`'F"u) /\ s = (`'F"v)))) -> Y e. A)
100		ssexg 3457	. . . . . . . . . . 11 |- (((`'F"(u i^i v)) C_ Y /\ Y e. A) -> (`'F"(u i^i v)) e. _V)
101	98, 99, 100	syl11anc 524	. . . . . . . . . 10 |- ((((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) /\ ((u e. L /\ v e. L) /\ (r = (`'F"u) /\ s = (`'F"v)))) -> (`'F"(u i^i v)) e. _V)
102		eqeq1 1890	. . . . . . . . . . . 12 |- (y = (`'F"(u i^i v)) -> (y = (`'F"x) <-> (`'F"(u i^i v)) = (`'F"x)))
103	102	rexbidv 2124	. . . . . . . . . . 11 |- (y = (`'F"(u i^i v)) -> (E.x e. L y = (`'F"x) <-> E.x e. L (`'F"(u i^i v)) = (`'F"x)))
104	103	elabg 2405	. . . . . . . . . 10 |- ((`'F"(u i^i v)) e. _V -> ((`'F"(u i^i v)) e. {y | E.x e. L y = (`'F"x)} <-> E.x e. L (`'F"(u i^i v)) = (`'F"x)))
105	101, 104	syl 12	. . . . . . . . 9 |- ((((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) /\ ((u e. L /\ v e. L) /\ (r = (`'F"u) /\ s = (`'F"v)))) -> ((`'F"(u i^i v)) e. {y | E.x e. L y = (`'F"x)} <-> E.x e. L (`'F"(u i^i v)) = (`'F"x)))
106	93, 105	mpbird 213	. . . . . . . 8 |- ((((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) /\ ((u e. L /\ v e. L) /\ (r = (`'F"u) /\ s = (`'F"v)))) -> (`'F"(u i^i v)) e. {y | E.x e. L y = (`'F"x)})
107		simprrl 458	. . . . . . . . . . 11 |- ((((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) /\ ((u e. L /\ v e. L) /\ (r = (`'F"u) /\ s = (`'F"v)))) -> r = (`'F"u))
108		simprrr 459	. . . . . . . . . . 11 |- ((((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) /\ ((u e. L /\ v e. L) /\ (r = (`'F"u) /\ s = (`'F"v)))) -> s = (`'F"v))
109	107, 108	ineq12d 2797	. . . . . . . . . 10 |- ((((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) /\ ((u e. L /\ v e. L) /\ (r = (`'F"u) /\ s = (`'F"v)))) -> (r i^i s) = ((`'F"u) i^i (`'F"v)))
110		funcnvcnv 4473	. . . . . . . . . . . . 13 |- (Fun F -> Fun `'`'F)
111		imain 4494	. . . . . . . . . . . . 13 |- (Fun `'`'F -> (`'F"(u i^i v)) = ((`'F"u) i^i (`'F"v)))
112	33, 110, 111	3syl 24	. . . . . . . . . . . 12 |- (F:Y-->X -> (`'F"(u i^i v)) = ((`'F"u) i^i (`'F"v)))
113	112	3ad2ant3 899	. . . . . . . . . . 11 |- ((Y e. A /\ L e. Fil /\ F:Y-->X) -> (`'F"(u i^i v)) = ((`'F"u) i^i (`'F"v)))
114	113	ad2antrr 440	. . . . . . . . . 10 |- ((((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) /\ ((u e. L /\ v e. L) /\ (r = (`'F"u) /\ s = (`'F"v)))) -> (`'F"(u i^i v)) = ((`'F"u) i^i (`'F"v)))
115	109, 114	eqtr4d 1928	. . . . . . . . 9 |- ((((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) /\ ((u e. L /\ v e. L) /\ (r = (`'F"u) /\ s = (`'F"v)))) -> (r i^i s) = (`'F"(u i^i v)))
116		eqimss2 2667	. . . . . . . . 9 |- ((r i^i s) = (`'F"(u i^i v)) -> (`'F"(u i^i v)) C_ (r i^i s))
117	115, 116	syl 12	. . . . . . . 8 |- ((((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) /\ ((u e. L /\ v e. L) /\ (r = (`'F"u) /\ s = (`'F"v)))) -> (`'F"(u i^i v)) C_ (r i^i s))
118		sseq1 2637	. . . . . . . . 9 |- (t = (`'F"(u i^i v)) -> (t C_ (r i^i s) <-> (`'F"(u i^i v)) C_ (r i^i s)))
119	118	rcla4ev 2381	. . . . . . . 8 |- (((`'F"(u i^i v)) e. {y | E.x e. L y = (`'F"x)} /\ (`'F"(u i^i v)) C_ (r i^i s)) -> E.t e. {y | E.x e. L y = (`'F"x)}t C_ (r i^i s))
120	106, 117, 119	syl11anc 524	. . . . . . 7 |- ((((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) /\ ((u e. L /\ v e. L) /\ (r = (`'F"u) /\ s = (`'F"v)))) -> E.t e. {y | E.x e. L y = (`'F"x)}t C_ (r i^i s))
121	120	exp32 408	. . . . . 6 |- (((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) -> ((u e. L /\ v e. L) -> ((r = (`'F"u) /\ s = (`'F"v)) -> E.t e. {y | E.x e. L y = (`'F"x)}t C_ (r i^i s))))
122	121	r19.23advv 2218	. . . . 5 |- (((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) -> (E.u e. L E.v e. L (r = (`'F"u) /\ s = (`'F"v)) -> E.t e. {y | E.x e. L y = (`'F"x)}t C_ (r i^i s)))
123		visset 2295	. . . . . . . . 9 |- r e. _V
124		eqeq1 1890	. . . . . . . . . 10 |- (y = r -> (y = (`'F"x) <-> r = (`'F"x)))
125	124	rexbidv 2124	. . . . . . . . 9 |- (y = r -> (E.x e. L y = (`'F"x) <-> E.x e. L r = (`'F"x)))
126	123, 125	elab 2403	. . . . . . . 8 |- (r e. {y | E.x e. L y = (`'F"x)} <-> E.x e. L r = (`'F"x))
127		imaeq2 4260	. . . . . . . . . 10 |- (x = u -> (`'F"x) = (`'F"u))
128	127	eqeq2d 1895	. . . . . . . . 9 |- (x = u -> (r = (`'F"x) <-> r = (`'F"u)))
129	128	cbvrexv 2281	. . . . . . . 8 |- (E.x e. L r = (`'F"x) <-> E.u e. L r = (`'F"u))
130	126, 129	bitri 190	. . . . . . 7 |- (r e. {y | E.x e. L y = (`'F"x)} <-> E.u e. L r = (`'F"u))
131		visset 2295	. . . . . . . . 9 |- s e. _V
132		eqeq1 1890	. . . . . . . . . 10 |- (y = s -> (y = (`'F"x) <-> s = (`'F"x)))
133	132	rexbidv 2124	. . . . . . . . 9 |- (y = s -> (E.x e. L y = (`'F"x) <-> E.x e. L s = (`'F"x)))
134	131, 133	elab 2403	. . . . . . . 8 |- (s e. {y | E.x e. L y = (`'F"x)} <-> E.x e. L s = (`'F"x))
135		imaeq2 4260	. . . . . . . . . 10 |- (x = v -> (`'F"x) = (`'F"v))
136	135	eqeq2d 1895	. . . . . . . . 9 |- (x = v -> (s = (`'F"x) <-> s = (`'F"v)))
137	136	cbvrexv 2281	. . . . . . . 8 |- (E.x e. L s = (`'F"x) <-> E.v e. L s = (`'F"v))
138	134, 137	bitri 190	. . . . . . 7 |- (s e. {y | E.x e. L y = (`'F"x)} <-> E.v e. L s = (`'F"v))
139	130, 138	anbi12i 540	. . . . . 6 |- ((r e. {y | E.x e. L y = (`'F"x)} /\ s e. {y | E.x e. L y = (`'F"x)}) <-> (E.u e. L r = (`'F"u) /\ E.v e. L s = (`'F"v)))
140		reeanv 2249	. . . . . 6 |- (E.u e. L E.v e. L (r = (`'F"u) /\ s = (`'F"v)) <-> (E.u e. L r = (`'F"u) /\ E.v e. L s = (`'F"v)))
141	139, 140	bitr4i 193	. . . . 5 |- ((r e. {y | E.x e. L y = (`'F"x)} /\ s e. {y | E.x e. L y = (`'F"x)}) <-> E.u e. L E.v e. L (r = (`'F"u) /\ s = (`'F"v)))
142	122, 141	syl5ib 223	. . . 4 |- (((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) -> ((r e. {y | E.x e. L y = (`'F"x)} /\ s e. {y | E.x e. L y = (`'F"x)}) -> E.t e. {y | E.x e. L y = (`'F"x)}t C_ (r i^i s)))
143	142	r19.21aivv 2183	. . 3 |- (((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) -> A.r e. {y | E.x e. L y = (`'F"x)}A.s e. {y | E.x e. L y = (`'F"x)}E.t e. {y | E.x e. L y = (`'F"x)}t C_ (r i^i s))
144	20, 83, 143	3jca 1050	. 2 |- (((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) -> ({y | E.x e. L y = (`'F"x)} =/= (/) /\ (/) e/ {y | E.x e. L y = (`'F"x)} /\ A.r e. {y | E.x e. L y = (`'F"x)}A.s e. {y | E.x e. L y = (`'F"x)}E.t e. {y | E.x e. L y = (`'F"x)}t C_ (r i^i s)))
145		abrexexg 4837	. . . . 5 |- (L e. Fil -> {y | E.x e. L y = (`'F"x)} e. _V)
146		isfbas2 10263	. . . . 5 |- ({y | E.x e. L y = (`'F"x)} e. _V -> ({y | E.x e. L y = (`'F"x)} e. fBas <-> ({y | E.x e. L y = (`'F"x)} =/= (/) /\ (/) e/ {y | E.x e. L y = (`'F"x)} /\ A.r e. {y | E.x e. L y = (`'F"x)}A.s e. {y | E.x e. L y = (`'F"x)}E.t e. {y | E.x e. L y = (`'F"x)}t C_ (r i^i s))))
147	145, 146	syl 12	. . . 4 |- (L e. Fil -> ({y | E.x e. L y = (`'F"x)} e. fBas <-> ({y | E.x e. L y = (`'F"x)} =/= (/) /\ (/) e/ {y | E.x e. L y = (`'F"x)} /\ A.r e. {y | E.x e. L y = (`'F"x)}A.s e. {y | E.x e. L y = (`'F"x)}E.t e. {y | E.x e. L y = (`'F"x)}t C_ (r i^i s))))
148	147	3ad2ant2 898	. . 3 |- ((Y e. A /\ L e. Fil /\ F:Y-->X) -> ({y | E.x e. L y = (`'F"x)} e. fBas <-> ({y | E.x e. L y = (`'F"x)} =/= (/) /\ (/) e/ {y | E.x e. L y = (`'F"x)} /\ A.r e. {y | E.x e. L y = (`'F"x)}A.s e. {y | E.x e. L y = (`'F"x)}E.t e. {y | E.x e. L y = (`'F"x)}t C_ (r i^i s))))
149	148	adantr 425	. 2 |- (((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) -> ({y | E.x e. L y = (`'F"x)} e. fBas <-> ({y | E.x e. L y = (`'F"x)} =/= (/) /\ (/) e/ {y | E.x e. L y = (`'F"x)} /\ A.r e. {y | E.x e. L y = (`'F"x)}A.s e. {y | E.x e. L y = (`'F"x)}E.t e. {y | E.x e. L y = (`'F"x)}t C_ (r i^i s))))
150	144, 149	mpbird 213	1 |- (((Y e. A /\ L e. Fil /\ F:Y-->X) /\ ran F e. L) -> {y | E.x e. L y = (`'F"x)} e. fBas)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858  A.wal 1296   = wceq 1298   e. wcel 1300  {cab 1871   =/= wne 2017   e/ wnel 2018  A.wral 2105  E.wrex 2106  _Vcvv 2292   i^i cin 2592   C_ wss 2593  (/)c0 2875  U.cuni 3177  `'ccnv 3985  dom cdm 3986  ran crn 3987  "cima 3989  Fun wfun 3992   Fn wfn 3993  -->wf 3994  ` cfv 3998  fBascfbas 10257  Filcfil 10264

This theorem is referenced by:  rnelfm 15593  fmfnfm 15598

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-rep 3428  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-3an 860  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-nel 2020  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  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-f 4010  df-fv 4014  df-fbas 10259  df-fil 10265

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