Theorem rankmapu 9567

Description: An upper bound on the rank of set exponentiation. (Contributed by Gérard Lang, 5-Aug-2018.)

Hypotheses

Ref	Expression
rankxpl.1	⊢ 𝐴 ∈ V
rankxpl.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
rankmapu	⊢ (rank‘(𝐴 ↑m 𝐵)) ⊆ suc suc suc (rank‘(𝐴 ∪ 𝐵))

Proof of Theorem rankmapu

Step	Hyp	Ref	Expression
1		mapsspw 8624	. . 3 ⊢ (𝐴 ↑m 𝐵) ⊆ 𝒫 (𝐵 × 𝐴)
2		rankxpl.2	. . . . . 6 ⊢ 𝐵 ∈ V
3		rankxpl.1	. . . . . 6 ⊢ 𝐴 ∈ V
4	2, 3	xpex 7581	. . . . 5 ⊢ (𝐵 × 𝐴) ∈ V
5	4	pwex 5298	. . . 4 ⊢ 𝒫 (𝐵 × 𝐴) ∈ V
6	5	rankss 9538	. . 3 ⊢ ((𝐴 ↑m 𝐵) ⊆ 𝒫 (𝐵 × 𝐴) → (rank‘(𝐴 ↑m 𝐵)) ⊆ (rank‘𝒫 (𝐵 × 𝐴)))
7	1, 6	ax-mp 5	. 2 ⊢ (rank‘(𝐴 ↑m 𝐵)) ⊆ (rank‘𝒫 (𝐵 × 𝐴))
8	4	rankpw 9532	. . 3 ⊢ (rank‘𝒫 (𝐵 × 𝐴)) = suc (rank‘(𝐵 × 𝐴))
9	2, 3	rankxpu 9565	. . . . 5 ⊢ (rank‘(𝐵 × 𝐴)) ⊆ suc suc (rank‘(𝐵 ∪ 𝐴))
10		uncom 4083	. . . . . . . 8 ⊢ (𝐵 ∪ 𝐴) = (𝐴 ∪ 𝐵)
11	10	fveq2i 6759	. . . . . . 7 ⊢ (rank‘(𝐵 ∪ 𝐴)) = (rank‘(𝐴 ∪ 𝐵))
12		suceq 6316	. . . . . . 7 ⊢ ((rank‘(𝐵 ∪ 𝐴)) = (rank‘(𝐴 ∪ 𝐵)) → suc (rank‘(𝐵 ∪ 𝐴)) = suc (rank‘(𝐴 ∪ 𝐵)))
13	11, 12	ax-mp 5	. . . . . 6 ⊢ suc (rank‘(𝐵 ∪ 𝐴)) = suc (rank‘(𝐴 ∪ 𝐵))
14		suceq 6316	. . . . . 6 ⊢ (suc (rank‘(𝐵 ∪ 𝐴)) = suc (rank‘(𝐴 ∪ 𝐵)) → suc suc (rank‘(𝐵 ∪ 𝐴)) = suc suc (rank‘(𝐴 ∪ 𝐵)))
15	13, 14	ax-mp 5	. . . . 5 ⊢ suc suc (rank‘(𝐵 ∪ 𝐴)) = suc suc (rank‘(𝐴 ∪ 𝐵))
16	9, 15	sseqtri 3953	. . . 4 ⊢ (rank‘(𝐵 × 𝐴)) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵))
17		rankon 9484	. . . . . 6 ⊢ (rank‘(𝐵 × 𝐴)) ∈ On
18	17	onordi 6356	. . . . 5 ⊢ Ord (rank‘(𝐵 × 𝐴))
19		rankon 9484	. . . . . . . 8 ⊢ (rank‘(𝐴 ∪ 𝐵)) ∈ On
20	19	onsuci 7660	. . . . . . 7 ⊢ suc (rank‘(𝐴 ∪ 𝐵)) ∈ On
21	20	onsuci 7660	. . . . . 6 ⊢ suc suc (rank‘(𝐴 ∪ 𝐵)) ∈ On
22	21	onordi 6356	. . . . 5 ⊢ Ord suc suc (rank‘(𝐴 ∪ 𝐵))
23		ordsucsssuc 7645	. . . . 5 ⊢ ((Ord (rank‘(𝐵 × 𝐴)) ∧ Ord suc suc (rank‘(𝐴 ∪ 𝐵))) → ((rank‘(𝐵 × 𝐴)) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵)) ↔ suc (rank‘(𝐵 × 𝐴)) ⊆ suc suc suc (rank‘(𝐴 ∪ 𝐵))))
24	18, 22, 23	mp2an 688	. . . 4 ⊢ ((rank‘(𝐵 × 𝐴)) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵)) ↔ suc (rank‘(𝐵 × 𝐴)) ⊆ suc suc suc (rank‘(𝐴 ∪ 𝐵)))
25	16, 24	mpbi 229	. . 3 ⊢ suc (rank‘(𝐵 × 𝐴)) ⊆ suc suc suc (rank‘(𝐴 ∪ 𝐵))
26	8, 25	eqsstri 3951	. 2 ⊢ (rank‘𝒫 (𝐵 × 𝐴)) ⊆ suc suc suc (rank‘(𝐴 ∪ 𝐵))
27	7, 26	sstri 3926	1 ⊢ (rank‘(𝐴 ↑m 𝐵)) ⊆ suc suc suc (rank‘(𝐴 ∪ 𝐵))

Syntax hints:   ↔ wb 205   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ∪ cun 3881   ⊆ wss 3883  𝒫 cpw 4530   × cxp 5578  Ord word 6250  suc csuc 6253  ‘cfv 6418  (class class class)co 7255   ↑_m cmap 8573  rankcrnk 9452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-reg 9281  ax-inf2 9329

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-map 8575  df-pm 8576  df-r1 9453  df-rank 9454

This theorem is referenced by: (None)