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Theorem rankelg 35764
Description: The membership relation is inherited by the rank function. Closed form of rankel 9862. (Contributed by Scott Fenton, 16-Jul-2015.)
Assertion
Ref Expression
rankelg ((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ 𝐡) β†’ (rankβ€˜π΄) ∈ (rankβ€˜π΅))

Proof of Theorem rankelg
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eleq2 2818 . . . 4 (𝑦 = 𝐡 β†’ (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝐡))
2 fveq2 6897 . . . . 5 (𝑦 = 𝐡 β†’ (rankβ€˜π‘¦) = (rankβ€˜π΅))
32eleq2d 2815 . . . 4 (𝑦 = 𝐡 β†’ ((rankβ€˜π΄) ∈ (rankβ€˜π‘¦) ↔ (rankβ€˜π΄) ∈ (rankβ€˜π΅)))
41, 3imbi12d 344 . . 3 (𝑦 = 𝐡 β†’ ((𝐴 ∈ 𝑦 β†’ (rankβ€˜π΄) ∈ (rankβ€˜π‘¦)) ↔ (𝐴 ∈ 𝐡 β†’ (rankβ€˜π΄) ∈ (rankβ€˜π΅))))
5 vex 3475 . . . 4 𝑦 ∈ V
65rankel 9862 . . 3 (𝐴 ∈ 𝑦 β†’ (rankβ€˜π΄) ∈ (rankβ€˜π‘¦))
74, 6vtoclg 3540 . 2 (𝐡 ∈ 𝑉 β†’ (𝐴 ∈ 𝐡 β†’ (rankβ€˜π΄) ∈ (rankβ€˜π΅)))
87imp 406 1 ((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ 𝐡) β†’ (rankβ€˜π΄) ∈ (rankβ€˜π΅))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  β€˜cfv 6548  rankcrnk 9786
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 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-reg 9615  ax-inf2 9664
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 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-om 7871  df-2nd 7994  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-r1 9787  df-rank 9788
This theorem is referenced by:  hfelhf  35777
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