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Theorem fncnv 4908
Description: Single-rootedness (see funcnv 4903) of a class cut down by a cross product. (Contributed by NM, 5-Mar-2007.)
Assertion
Ref Expression
fncnv  `' R  i^i  X.  Fn  R
Distinct variable groups:   ,,   ,,   , R,

Proof of Theorem fncnv
StepHypRef Expression
1 df-fn 4848 . 2  `' R  i^i  X.  Fn  Fun  `' R  i^i  X.  dom  `' R  i^i  X.
2 df-rn 4299 . . . 4  ran  R  i^i  X.  dom  `' R  i^i  X.
32eqeq1i 2044 . . 3  ran  R  i^i  X.  dom  `' R  i^i  X.
43anbi2i 430 . 2  Fun  `' R  i^i  X.  ran  R  i^i  X.  Fun  `' R  i^i  X.  dom  `' R  i^i  X.
5 rninxp 4707 . . . . 5  ran  R  i^i  X.  R
65anbi1i 431 . . . 4  ran  R  i^i  X.  R  R  R
7 funcnv 4903 . . . . . 6  Fun  `' R  i^i  X.  ran  R  i^i  X.  R  i^i  X.
8 raleq 2499 . . . . . . 7  ran  R  i^i  X.  ran  R  i^i  X.  R  i^i  X.  R  i^i  X.
9 biimt 230 . . . . . . . . 9  R  R
10 moanimv 1972 . . . . . . . . . 10  R  R
11 brinxp2 4350 . . . . . . . . . . . 12  R  i^i  X.  R
12 3anan12 896 . . . . . . . . . . . 12  R  R
1311, 12bitri 173 . . . . . . . . . . 11  R  i^i  X.  R
1413mobii 1934 . . . . . . . . . 10  R  i^i  X.  R
15 df-rmo 2308 . . . . . . . . . . 11  R  R
1615imbi2i 215 . . . . . . . . . 10  R  R
1710, 14, 163bitr4i 201 . . . . . . . . 9  R  i^i  X.  R
189, 17syl6rbbr 188 . . . . . . . 8  R  i^i  X.  R
1918ralbiia 2332 . . . . . . 7  R  i^i  X.  R
208, 19syl6bb 185 . . . . . 6  ran  R  i^i  X.  ran  R  i^i  X.  R  i^i  X.  R
217, 20syl5bb 181 . . . . 5  ran  R  i^i  X.  Fun  `' R  i^i  X.  R
2221pm5.32i 427 . . . 4  ran  R  i^i  X.  Fun  `' R  i^i  X.  ran  R  i^i  X.  R
23 r19.26 2435 . . . 4  R  R  R  R
246, 22, 233bitr4i 201 . . 3  ran  R  i^i  X.  Fun  `' R  i^i  X.  R  R
25 ancom 253 . . 3  Fun  `' R  i^i  X.  ran  R  i^i  X.  ran  R  i^i  X.  Fun  `' R  i^i  X.
26 reu5 2516 . . . 4  R  R  R
2726ralbii 2324 . . 3  R  R  R
2824, 25, 273bitr4i 201 . 2  Fun  `' R  i^i  X.  ran  R  i^i  X.  R
291, 4, 283bitr2i 197 1  `' R  i^i  X.  Fn  R
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   w3a 884   wceq 1242   wcel 1390  wmo 1898  wral 2300  wrex 2301  wreu 2302  wrmo 2303    i^i cin 2910   class class class wbr 3755    X. cxp 4286   `'ccnv 4287   dom cdm 4288   ran crn 4289   Fun wfun 4839    Fn wfn 4840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-rmo 2308  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-fun 4847  df-fn 4848
This theorem is referenced by: (None)
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