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Theorem r1ord 8148
Description: Ordering relation for the cumulative hierarchy of sets. Part of Proposition 9.10(2) of [TakeutiZaring] p. 77. (Contributed by NM, 8-Sep-2003.) (Revised by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
r1ord  |-  ( B  e.  On  ->  ( A  e.  B  ->  ( R1 `  A )  e.  ( R1 `  B ) ) )

Proof of Theorem r1ord
StepHypRef Expression
1 r1fnon 8135 . . . 4  |-  R1  Fn  On
2 fndm 5615 . . . 4  |-  ( R1  Fn  On  ->  dom  R1  =  On )
31, 2ax-mp 5 . . 3  |-  dom  R1  =  On
43eleq2i 2478 . 2  |-  ( B  e.  dom  R1  <->  B  e.  On )
5 r1ordg 8146 . 2  |-  ( B  e.  dom  R1  ->  ( A  e.  B  -> 
( R1 `  A
)  e.  ( R1
`  B ) ) )
64, 5sylbir 213 1  |-  ( B  e.  On  ->  ( A  e.  B  ->  ( R1 `  A )  e.  ( R1 `  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1403    e. wcel 1840   Oncon0 4819   dom cdm 4940    Fn wfn 5518   ` cfv 5523   R1cr1 8130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-om 6637  df-recs 6997  df-rdg 7031  df-r1 8132
This theorem is referenced by:  r1ord2  8149  r1wunlim  9063
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