Theorem rankmapu 9831

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 8851	. . 3 ⊢ (𝐴 ↑m 𝐵) ⊆ 𝒫 (𝐵 × 𝐴)
2		rankxpl.2	. . . . . 6 ⊢ 𝐵 ∈ V
3		rankxpl.1	. . . . . 6 ⊢ 𝐴 ∈ V
4	2, 3	xpex 7729	. . . . 5 ⊢ (𝐵 × 𝐴) ∈ V
5	4	pwex 5335	. . . 4 ⊢ 𝒫 (𝐵 × 𝐴) ∈ V
6	5	rankss 9802	. . 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 9796	. . 3 ⊢ (rank‘𝒫 (𝐵 × 𝐴)) = suc (rank‘(𝐵 × 𝐴))
9	2, 3	rankxpu 9829	. . . . 5 ⊢ (rank‘(𝐵 × 𝐴)) ⊆ suc suc (rank‘(𝐵 ∪ 𝐴))
10		uncom 4121	. . . . . . . 8 ⊢ (𝐵 ∪ 𝐴) = (𝐴 ∪ 𝐵)
11	10	fveq2i 6861	. . . . . . 7 ⊢ (rank‘(𝐵 ∪ 𝐴)) = (rank‘(𝐴 ∪ 𝐵))
12		suceq 6400	. . . . . . 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 6400	. . . . . 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 3995	. . . 4 ⊢ (rank‘(𝐵 × 𝐴)) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵))
17		rankon 9748	. . . . . 6 ⊢ (rank‘(𝐵 × 𝐴)) ∈ On
18	17	onordi 6445	. . . . 5 ⊢ Ord (rank‘(𝐵 × 𝐴))
19		rankon 9748	. . . . . . . 8 ⊢ (rank‘(𝐴 ∪ 𝐵)) ∈ On
20	19	onsuci 7814	. . . . . . 7 ⊢ suc (rank‘(𝐴 ∪ 𝐵)) ∈ On
21	20	onsuci 7814	. . . . . 6 ⊢ suc suc (rank‘(𝐴 ∪ 𝐵)) ∈ On
22	21	onordi 6445	. . . . 5 ⊢ Ord suc suc (rank‘(𝐴 ∪ 𝐵))
23		ordsucsssuc 7798	. . . . 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 692	. . . 4 ⊢ ((rank‘(𝐵 × 𝐴)) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵)) ↔ suc (rank‘(𝐵 × 𝐴)) ⊆ suc suc suc (rank‘(𝐴 ∪ 𝐵)))
25	16, 24	mpbi 230	. . 3 ⊢ suc (rank‘(𝐵 × 𝐴)) ⊆ suc suc suc (rank‘(𝐴 ∪ 𝐵))
26	8, 25	eqsstri 3993	. 2 ⊢ (rank‘𝒫 (𝐵 × 𝐴)) ⊆ suc suc suc (rank‘(𝐴 ∪ 𝐵))
27	7, 26	sstri 3956	1 ⊢ (rank‘(𝐴 ↑m 𝐵)) ⊆ suc suc suc (rank‘(𝐴 ∪ 𝐵))

Syntax hints:   ↔ wb 206   = wceq 1540   ∈ wcel 2109  Vcvv 3447   ∪ cun 3912   ⊆ wss 3914  𝒫 cpw 4563   × cxp 5636  Ord word 6331  suc csuc 6334  ‘cfv 6511  (class class class)co 7387   ↑_m cmap 8799  rankcrnk 9716

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-reg 9545  ax-inf2 9594

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-map 8801  df-pm 8802  df-r1 9717  df-rank 9718

This theorem is referenced by: (None)