Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rankelg Structured version   Visualization version   GIF version

Theorem rankelg 35700
Description: The membership relation is inherited by the rank function. Closed form of rankel 9854. (Contributed by Scott Fenton, 16-Jul-2015.)
Assertion
Ref Expression
rankelg ((𝐵𝑉𝐴𝐵) → (rank‘𝐴) ∈ (rank‘𝐵))

Proof of Theorem rankelg
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eleq2 2817 . . . 4 (𝑦 = 𝐵 → (𝐴𝑦𝐴𝐵))
2 fveq2 6891 . . . . 5 (𝑦 = 𝐵 → (rank‘𝑦) = (rank‘𝐵))
32eleq2d 2814 . . . 4 (𝑦 = 𝐵 → ((rank‘𝐴) ∈ (rank‘𝑦) ↔ (rank‘𝐴) ∈ (rank‘𝐵)))
41, 3imbi12d 344 . . 3 (𝑦 = 𝐵 → ((𝐴𝑦 → (rank‘𝐴) ∈ (rank‘𝑦)) ↔ (𝐴𝐵 → (rank‘𝐴) ∈ (rank‘𝐵))))
5 vex 3473 . . . 4 𝑦 ∈ V
65rankel 9854 . . 3 (𝐴𝑦 → (rank‘𝐴) ∈ (rank‘𝑦))
74, 6vtoclg 3538 . 2 (𝐵𝑉 → (𝐴𝐵 → (rank‘𝐴) ∈ (rank‘𝐵)))
87imp 406 1 ((𝐵𝑉𝐴𝐵) → (rank‘𝐴) ∈ (rank‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  cfv 6542  rankcrnk 9778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-reg 9607  ax-inf2 9656
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-om 7865  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-r1 9779  df-rank 9780
This theorem is referenced by:  hfelhf  35713
  Copyright terms: Public domain W3C validator