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Theorem rankvalb 7683
 Description: Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 7702 does not use Regularity, and so requires the assumption that is in the range of . (Contributed by NM, 11-Oct-2003.) (Revised by Mario Carneiro, 10-Sep-2013.)
Assertion
Ref Expression
rankvalb
Distinct variable group:   ,

Proof of Theorem rankvalb
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elex 2928 . 2
2 rankwflemb 7679 . . . 4
3 intexrab 4323 . . . 4
42, 3bitri 241 . . 3
54biimpi 187 . 2
6 eleq1 2468 . . . . 5
76rabbidv 2912 . . . 4
87inteqd 4019 . . 3
9 df-rank 7651 . . 3
108, 9fvmptg 5767 . 2
111, 5, 10syl2anc 643 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1649   wcel 1721  wrex 2671  crab 2674  cvv 2920  cuni 3979  cint 4014  con0 4545   csuc 4547  cima 4844  cfv 5417  cr1 7648  crnk 7649 This theorem is referenced by:  rankr1ai  7684  rankidb  7686  rankval  7702 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-recs 6596  df-rdg 6631  df-r1 7650  df-rank 7651
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