Theorem rankmapu 9895

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 8906	. . 3 ⊢ (𝐴 ↑m 𝐵) ⊆ 𝒫 (𝐵 × 𝐴)
2		rankxpl.2	. . . . . 6 ⊢ 𝐵 ∈ V
3		rankxpl.1	. . . . . 6 ⊢ 𝐴 ∈ V
4	2, 3	xpex 7767	. . . . 5 ⊢ (𝐵 × 𝐴) ∈ V
5	4	pwex 5341	. . . 4 ⊢ 𝒫 (𝐵 × 𝐴) ∈ V
6	5	rankss 9863	. . 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 9856	. . 3 ⊢ (rank‘𝒫 (𝐵 × 𝐴)) = suc (rank‘(𝐵 × 𝐴))
9	2, 3	rankxpu 9893	. . . . 5 ⊢ (rank‘(𝐵 × 𝐴)) ⊆ suc suc (rank‘(𝐵 ∪ 𝐴))
10		uncom 4157	. . . . . . . 8 ⊢ (𝐵 ∪ 𝐴) = (𝐴 ∪ 𝐵)
11	10	fveq2i 6888	. . . . . . 7 ⊢ (rank‘(𝐵 ∪ 𝐴)) = (rank‘(𝐴 ∪ 𝐵))
12		suceq 6431	. . . . . . 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 6431	. . . . . 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 3956	. . . 4 ⊢ (rank‘(𝐵 × 𝐴)) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵))
17		rankon 9803	. . . . . 6 ⊢ (rank‘(𝐵 × 𝐴)) ∈ On
18	17	onordi 6476	. . . . 5 ⊢ Ord (rank‘(𝐵 × 𝐴))
19		rankon 9803	. . . . . . . 8 ⊢ (rank‘(𝐴 ∪ 𝐵)) ∈ On
20	19	onsuci 7850	. . . . . . 7 ⊢ suc (rank‘(𝐴 ∪ 𝐵)) ∈ On
21	20	onsuci 7850	. . . . . 6 ⊢ suc suc (rank‘(𝐴 ∪ 𝐵)) ∈ On
22	21	onordi 6476	. . . . 5 ⊢ Ord suc suc (rank‘(𝐴 ∪ 𝐵))
23		ordsucsssuc 7834	. . . . 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 705	. . . 4 ⊢ ((rank‘(𝐵 × 𝐴)) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵)) ↔ suc (rank‘(𝐵 × 𝐴)) ⊆ suc suc suc (rank‘(𝐴 ∪ 𝐵)))
25	16, 24	mpbi 233	. . 3 ⊢ suc (rank‘(𝐵 × 𝐴)) ⊆ suc suc suc (rank‘(𝐴 ∪ 𝐵))
26	8, 25	eqsstri 3954	. 2 ⊢ (rank‘𝒫 (𝐵 × 𝐴)) ⊆ suc suc suc (rank‘(𝐴 ∪ 𝐵))
27	7, 26	sstri 3917	1 ⊢ (rank‘(𝐴 ↑m 𝐵)) ⊆ suc suc suc (rank‘(𝐴 ∪ 𝐵))

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1569   ∈ wcel 2144  Vcvv 3450   ⊆ wss 3895   ∪ cun 4149  𝒫 cpw 4556   × cxp 5648  Ord word 6361  suc csuc 6364  ‘cfv 6538  (class class class)co 7420   ↑_m cmap 8847  rankcrnk 9767

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-reg 9586  ax-inf2 9642

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-map 8849  df-pm 8850  df-r1 9768  df-rank 9769

This theorem is used by: (None)