Theorem indexdom 15754

Description: If for every element of an indexing set A there exists a corresponding element of another set B, then there exists a subset of B consisting only of those elements which are indexed by A, and which is dominated by the set A.

Assertion

Ref	Expression
indexdom	|- ((A e. M /\ A.x e. A E.y e. B ph) -> E.c((c ~<_ A /\ c C_ B) /\ (A.x e. A E.y e. c ph /\ A.y e. c E.x e. A ph)))

Distinct variable groups:   A,c,x,y   B,c,x,y   ph,c

Proof of Theorem indexdom

Step	Hyp	Ref	Expression
1		fvex 4689	. . . 4 |- (f` x) e. _V
2	1	hbsbc1v 2464	. . 3 |- ([(f` x) / y]ph -> A.y[(f` x) / y]ph)
3		sbceq1a 2456	. . 3 |- (y = (f` x) -> (ph <-> [(f` x) / y]ph))
4	2, 3	ac6gf 15749	. 2 |- ((A e. M /\ A.x e. A E.y e. B ph) -> E.f(f:A-->B /\ A.x e. A [(f` x) / y]ph))
5		fdm 4567	. . . . . . 7 |- (f:A-->B -> dom f = A)
6		visset 2295	. . . . . . . 8 |- f e. _V
7	6	dmex 4208	. . . . . . 7 |- dom f e. _V
8	5, 7	syl6eqelr 1980	. . . . . 6 |- (f:A-->B -> A e. _V)
9		ffn 4562	. . . . . 6 |- (f:A-->B -> f Fn A)
10		fnrndomg 5969	. . . . . 6 |- (A e. _V -> (f Fn A -> ran f ~<_ A))
11	8, 9, 10	sylc 83	. . . . 5 |- (f:A-->B -> ran f ~<_ A)
12	11	adantr 425	. . . 4 |- ((f:A-->B /\ A.x e. A [(f` x) / y]ph) -> ran f ~<_ A)
13		frn 4569	. . . . 5 |- (f:A-->B -> ran f C_ B)
14	13	adantr 425	. . . 4 |- ((f:A-->B /\ A.x e. A [(f` x) / y]ph) -> ran f C_ B)
15		ax-17 1317	. . . . . 6 |- (f:A-->B -> A.x f:A-->B)
16		hbra1 2147	. . . . . 6 |- (A.x e. A [(f` x) / y]ph -> A.xA.x e. A [(f` x) / y]ph)
17	15, 16	hban 1356	. . . . 5 |- ((f:A-->B /\ A.x e. A [(f` x) / y]ph) -> A.x(f:A-->B /\ A.x e. A [(f` x) / y]ph))
18		ffun 4565	. . . . . . . . . 10 |- (f:A-->B -> Fun f)
19	18	adantr 425	. . . . . . . . 9 |- ((f:A-->B /\ x e. A) -> Fun f)
20	5	eleq2d 1964	. . . . . . . . . 10 |- (f:A-->B -> (x e. dom f <-> x e. A))
21	20	biimpar 461	. . . . . . . . 9 |- ((f:A-->B /\ x e. A) -> x e. dom f)
22		fvelrn 4785	. . . . . . . . 9 |- ((Fun f /\ x e. dom f) -> (f` x) e. ran f)
23	19, 21, 22	syl11anc 524	. . . . . . . 8 |- ((f:A-->B /\ x e. A) -> (f` x) e. ran f)
24	23	adantlr 429	. . . . . . 7 |- (((f:A-->B /\ A.x e. A [(f` x) / y]ph) /\ x e. A) -> (f` x) e. ran f)
25		ra4 2155	. . . . . . . . 9 |- (A.x e. A [(f` x) / y]ph -> (x e. A -> [(f` x) / y]ph))
26	25	imp 377	. . . . . . . 8 |- ((A.x e. A [(f` x) / y]ph /\ x e. A) -> [(f` x) / y]ph)
27	26	adantll 428	. . . . . . 7 |- (((f:A-->B /\ A.x e. A [(f` x) / y]ph) /\ x e. A) -> [(f` x) / y]ph)
28	2, 3	rcla4e 2375	. . . . . . 7 |- (((f` x) e. ran f /\ [(f` x) / y]ph) -> E.y e. ran fph)
29	24, 27, 28	syl11anc 524	. . . . . 6 |- (((f:A-->B /\ A.x e. A [(f` x) / y]ph) /\ x e. A) -> E.y e. ran fph)
30	29	ex 402	. . . . 5 |- ((f:A-->B /\ A.x e. A [(f` x) / y]ph) -> (x e. A -> E.y e. ran fph))
31	17, 30	r19.21ai 2174	. . . 4 |- ((f:A-->B /\ A.x e. A [(f` x) / y]ph) -> A.x e. A E.y e. ran fph)
32		ax-17 1317	. . . . . 6 |- (f:A-->B -> A.y f:A-->B)
33		ax-17 1317	. . . . . . 7 |- (x e. A -> A.y x e. A)
34	33, 2	hbral 2146	. . . . . 6 |- (A.x e. A [(f` x) / y]ph -> A.yA.x e. A [(f` x) / y]ph)
35	32, 34	hban 1356	. . . . 5 |- ((f:A-->B /\ A.x e. A [(f` x) / y]ph) -> A.y(f:A-->B /\ A.x e. A [(f` x) / y]ph))
36		fvelrnb 4719	. . . . . . . 8 |- (f Fn A -> (y e. ran f <-> E.x e. A (f` x) = y))
37	9, 36	syl 12	. . . . . . 7 |- (f:A-->B -> (y e. ran f <-> E.x e. A (f` x) = y))
38	37	adantr 425	. . . . . 6 |- ((f:A-->B /\ A.x e. A [(f` x) / y]ph) -> (y e. ran f <-> E.x e. A (f` x) = y))
39	25	adantl 424	. . . . . . . 8 |- ((f:A-->B /\ A.x e. A [(f` x) / y]ph) -> (x e. A -> [(f` x) / y]ph))
40	3	eqcoms 1887	. . . . . . . . 9 |- ((f` x) = y -> (ph <-> [(f` x) / y]ph))
41	40	biimprcd 173	. . . . . . . 8 |- ([(f` x) / y]ph -> ((f` x) = y -> ph))
42	39, 41	syl6 25	. . . . . . 7 |- ((f:A-->B /\ A.x e. A [(f` x) / y]ph) -> (x e. A -> ((f` x) = y -> ph)))
43	17, 42	reximdai 2199	. . . . . 6 |- ((f:A-->B /\ A.x e. A [(f` x) / y]ph) -> (E.x e. A (f` x) = y -> E.x e. A ph))
44	38, 43	sylbid 220	. . . . 5 |- ((f:A-->B /\ A.x e. A [(f` x) / y]ph) -> (y e. ran f -> E.x e. A ph))
45	35, 44	r19.21ai 2174	. . . 4 |- ((f:A-->B /\ A.x e. A [(f` x) / y]ph) -> A.y e. ran fE.x e. A ph)
46	6	rnex 4209	. . . . 5 |- ran f e. _V
47		breq1 3341	. . . . . . 7 |- (c = ran f -> (c ~<_ A <-> ran f ~<_ A))
48		sseq1 2637	. . . . . . 7 |- (c = ran f -> (c C_ B <-> ran f C_ B))
49	47, 48	anbi12d 690	. . . . . 6 |- (c = ran f -> ((c ~<_ A /\ c C_ B) <-> (ran f ~<_ A /\ ran f C_ B)))
50		rexeq 2267	. . . . . . . 8 |- (c = ran f -> (E.y e. c ph <-> E.y e. ran fph))
51	50	ralbidv 2123	. . . . . . 7 |- (c = ran f -> (A.x e. A E.y e. c ph <-> A.x e. A E.y e. ran fph))
52		raleq 2266	. . . . . . 7 |- (c = ran f -> (A.y e. c E.x e. A ph <-> A.y e. ran fE.x e. A ph))
53	51, 52	anbi12d 690	. . . . . 6 |- (c = ran f -> ((A.x e. A E.y e. c ph /\ A.y e. c E.x e. A ph) <-> (A.x e. A E.y e. ran fph /\ A.y e. ran fE.x e. A ph)))
54	49, 53	anbi12d 690	. . . . 5 |- (c = ran f -> (((c ~<_ A /\ c C_ B) /\ (A.x e. A E.y e. c ph /\ A.y e. c E.x e. A ph)) <-> ((ran f ~<_ A /\ ran f C_ B) /\ (A.x e. A E.y e. ran fph /\ A.y e. ran fE.x e. A ph))))
55	46, 54	cla4ev 2371	. . . 4 |- (((ran f ~<_ A /\ ran f C_ B) /\ (A.x e. A E.y e. ran fph /\ A.y e. ran fE.x e. A ph)) -> E.c((c ~<_ A /\ c C_ B) /\ (A.x e. A E.y e. c ph /\ A.y e. c E.x e. A ph)))
56	12, 14, 31, 45, 55	syl22anc 1101	. . 3 |- ((f:A-->B /\ A.x e. A [(f` x) / y]ph) -> E.c((c ~<_ A /\ c C_ B) /\ (A.x e. A E.y e. c ph /\ A.y e. c E.x e. A ph)))
57	56	19.23aiv 1674	. 2 |- (E.f(f:A-->B /\ A.x e. A [(f` x) / y]ph) -> E.c((c ~<_ A /\ c C_ B) /\ (A.x e. A E.y e. c ph /\ A.y e. c E.x e. A ph)))
58	4, 57	syl 12	1 |- ((A e. M /\ A.x e. A E.y e. B ph) -> E.c((c ~<_ A /\ c C_ B) /\ (A.x e. A E.y e. c ph /\ A.y e. c E.x e. A ph)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  [wsbc 1534  A.wral 2105  E.wrex 2106  _Vcvv 2292   C_ wss 2593   class class class wbr 3338  dom cdm 3986  ran crn 3987  Fun wfun 3992   Fn wfn 3993  -->wf 3994  ` cfv 3998   ~<_ cdom 5424

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  ax-reg 5695  ax-inf2 5731  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  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-ral 2109  df-rex 2110  df-reu 2111  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-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-iin 3258  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  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-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-rdg 5140  df-en 5427  df-dom 5428  df-r1 5750  df-rank 5751

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