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Theorem rankelg 28211
Description: The membership relation is inherited by the rank function. Closed form of rankel 8051. (Contributed by Scott Fenton, 16-Jul-2015.)
Assertion
Ref Expression
rankelg  |-  ( ( B  e.  V  /\  A  e.  B )  ->  ( rank `  A
)  e.  ( rank `  B ) )

Proof of Theorem rankelg
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eleq2 2504 . . . 4  |-  ( y  =  B  ->  ( A  e.  y  <->  A  e.  B ) )
2 fveq2 5696 . . . . 5  |-  ( y  =  B  ->  ( rank `  y )  =  ( rank `  B
) )
32eleq2d 2510 . . . 4  |-  ( y  =  B  ->  (
( rank `  A )  e.  ( rank `  y
)  <->  ( rank `  A
)  e.  ( rank `  B ) ) )
41, 3imbi12d 320 . . 3  |-  ( y  =  B  ->  (
( A  e.  y  ->  ( rank `  A
)  e.  ( rank `  y ) )  <->  ( A  e.  B  ->  ( rank `  A )  e.  (
rank `  B )
) ) )
5 vex 2980 . . . 4  |-  y  e. 
_V
65rankel 8051 . . 3  |-  ( A  e.  y  ->  ( rank `  A )  e.  ( rank `  y
) )
74, 6vtoclg 3035 . 2  |-  ( B  e.  V  ->  ( A  e.  B  ->  (
rank `  A )  e.  ( rank `  B
) ) )
87imp 429 1  |-  ( ( B  e.  V  /\  A  e.  B )  ->  ( rank `  A
)  e.  ( rank `  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   ` cfv 5423   rankcrnk 7975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-reg 7812  ax-inf2 7852
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-om 6482  df-recs 6837  df-rdg 6871  df-r1 7976  df-rank 7977
This theorem is referenced by:  hfelhf  28224
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