Theorem fssxp 3628

Description: A mapping is a class of ordered pairs.

Assertion

Ref	Expression
fssxp	|- (F:A-->B -> F (_ (A X. B))

Proof of Theorem fssxp

Step	Hyp	Ref	Expression
1		frn 3624	. . . 4 |- (F:A-->B -> ran F (_ B)
2		ssid 2076	. . . . 5 |- A (_ A
3		ssxp 3251	. . . . 5 |- ((A (_ A /\ ran F (_ B) -> (A X. ran F) (_ (A X. B))
4	2, 3	mpan 694	. . . 4 |- (ran F (_ B -> (A X. ran F) (_ (A X. B))
5	1, 4	syl 10	. . 3 |- (F:A-->B -> (A X. ran F) (_ (A X. B))
6		fdm 3623	. . . 4 |- (F:A-->B -> dom F = A)
7		xpeq1 3195	. . . 4 |- (dom F = A -> (dom F X. ran F) = (A X. ran F))
8		sseq1 2078	. . . 4 |- ((dom F X. ran F) = (A X. ran F) -> ((dom F X. ran F) (_ (A X. B) <-> (A X. ran F) (_ (A X. B)))
9	6, 7, 8	3syl 20	. . 3 |- (F:A-->B -> ((dom F X. ran F) (_ (A X. B) <-> (A X. ran F) (_ (A X. B)))
10	5, 9	mpbird 196	. 2 |- (F:A-->B -> (dom F X. ran F) (_ (A X. B))
11		frel 3622	. . 3 |- (F:A-->B -> Rel F)
12		relssdr 3505	. . 3 |- (Rel F -> F (_ (dom F X. ran F))
13		sstr2 2067	. . 3 |- (F (_ (dom F X. ran F) -> ((dom F X. ran F) (_ (A X. B) -> F (_ (A X. B)))
14	11, 12, 13	3syl 20	. 2 |- (F:A-->B -> ((dom F X. ran F) (_ (A X. B) -> F (_ (A X. B)))
15	10, 14	mpd 26	1 |- (F:A-->B -> F (_ (A X. B))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 954   (_ wss 2043   X. cxp 3163  dom cdm 3165  ran crn 3166  Rel wrel 3170  -->wf 3173

This theorem is referenced by:  funssxp 3629  opelf 3631  fabexg 3644  dff4 3807  dff2 3808  fopabssxp 3815  mapex 4318  mapval2 4325  mapsspw 4331  uniixp 4347  infmap2 7531  lmbrf 7882  iscauf 7891  iscau5 7893  lmclimnn 7915  h2hcau 8788  h2hlm 8789  1alg 10534

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662  df-xp 3179  df-rel 3180  df-cnv 3181  df-dm 3183  df-rn 3184  df-fun 3187  df-fn 3188  df-f 3189

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