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Theorem resflem 5550
 Description: A lemma to bound the range of a restriction. The conclusion would also hold with in place of (provided does not occur in ). If that stronger result is needed, it is however simpler to use the instance of resflem 5550 where is substituted for (in both the conclusion and the third hypothesis). (Contributed by BJ, 4-Jul-2022.)
Hypotheses
Ref Expression
resflem.1
resflem.2
resflem.3
Assertion
Ref Expression
resflem
Distinct variable groups:   ,   ,   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem resflem
StepHypRef Expression
1 resflem.2 . . . . . 6
21sseld 3064 . . . . 5
3 resflem.1 . . . . . . 7
4 fdm 5246 . . . . . . 7
53, 4syl 14 . . . . . 6
65eleq2d 2185 . . . . 5
72, 6sylibrd 168 . . . 4
8 resflem.3 . . . . 5
98ex 114 . . . 4
107, 9jcad 303 . . 3
1110ralrimiv 2479 . 2
12 ffun 5243 . . . 4
133, 12syl 14 . . 3
14 ffvresb 5549 . . 3
1513, 14syl 14 . 2
1611, 15mpbird 166 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wb 104   wceq 1314   wcel 1463  wral 2391   wss 3039   cdm 4507   cres 4509   wfun 5085  wf 5087  cfv 5091 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099 This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-fv 5099 This theorem is referenced by: (None)
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