Theorem rankpwg 32815

Description: The rank of a power set. Closed form of rankpw 8983. (Contributed by Scott Fenton, 16-Jul-2015.)

Assertion

Ref	Expression
rankpwg	⊢ (𝐴 ∈ 𝑉 → (rank‘𝒫 𝐴) = suc (rank‘𝐴))

Proof of Theorem rankpwg

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pweq 4381	. . . 4 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
2	1	fveq2d 6437	. . 3 ⊢ (𝑥 = 𝐴 → (rank‘𝒫 𝑥) = (rank‘𝒫 𝐴))
3		fveq2 6433	. . . 4 ⊢ (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴))
4		suceq 6028	. . . 4 ⊢ ((rank‘𝑥) = (rank‘𝐴) → suc (rank‘𝑥) = suc (rank‘𝐴))
5	3, 4	syl 17	. . 3 ⊢ (𝑥 = 𝐴 → suc (rank‘𝑥) = suc (rank‘𝐴))
6	2, 5	eqeq12d 2840	. 2 ⊢ (𝑥 = 𝐴 → ((rank‘𝒫 𝑥) = suc (rank‘𝑥) ↔ (rank‘𝒫 𝐴) = suc (rank‘𝐴)))
7		vex 3417	. . 3 ⊢ 𝑥 ∈ V
8	7	rankpw 8983	. 2 ⊢ (rank‘𝒫 𝑥) = suc (rank‘𝑥)
9	6, 8	vtoclg 3482	1 ⊢ (𝐴 ∈ 𝑉 → (rank‘𝒫 𝐴) = suc (rank‘𝐴))

Syntax hints:   → wi 4   = wceq 1658   ∈ wcel 2166  𝒫 cpw 4378  suc csuc 5965  ‘cfv 6123  rankcrnk 8903

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-reg 8766  ax-inf2 8815

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-om 7327  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-r1 8904  df-rank 8905

This theorem is referenced by:  hfpw  32831