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Theorem rankelg 32612
Description: The membership relation is inherited by the rank function. Closed form of rankel 8866. (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 2839 . . . 4 (𝑦 = 𝐵 → (𝐴𝑦𝐴𝐵))
2 fveq2 6332 . . . . 5 (𝑦 = 𝐵 → (rank‘𝑦) = (rank‘𝐵))
32eleq2d 2836 . . . 4 (𝑦 = 𝐵 → ((rank‘𝐴) ∈ (rank‘𝑦) ↔ (rank‘𝐴) ∈ (rank‘𝐵)))
41, 3imbi12d 333 . . 3 (𝑦 = 𝐵 → ((𝐴𝑦 → (rank‘𝐴) ∈ (rank‘𝑦)) ↔ (𝐴𝐵 → (rank‘𝐴) ∈ (rank‘𝐵))))
5 vex 3354 . . . 4 𝑦 ∈ V
65rankel 8866 . . 3 (𝐴𝑦 → (rank‘𝐴) ∈ (rank‘𝑦))
74, 6vtoclg 3417 . 2 (𝐵𝑉 → (𝐴𝐵 → (rank‘𝐴) ∈ (rank‘𝐵)))
87imp 393 1 ((𝐵𝑉𝐴𝐵) → (rank‘𝐴) ∈ (rank‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  cfv 6031  rankcrnk 8790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-reg 8653  ax-inf2 8702
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-r1 8791  df-rank 8792
This theorem is referenced by:  hfelhf  32625
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