Theorem rankmapu 9802

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 8828	. . 3 ⊢ (𝐴 ↑m 𝐵) ⊆ 𝒫 (𝐵 × 𝐴)
2		rankxpl.2	. . . . . 6 ⊢ 𝐵 ∈ V
3		rankxpl.1	. . . . . 6 ⊢ 𝐴 ∈ V
4	2, 3	xpex 7708	. . . . 5 ⊢ (𝐵 × 𝐴) ∈ V
5	4	pwex 5327	. . . 4 ⊢ 𝒫 (𝐵 × 𝐴) ∈ V
6	5	rankss 9773	. . 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 9767	. . 3 ⊢ (rank‘𝒫 (𝐵 × 𝐴)) = suc (rank‘(𝐵 × 𝐴))
9	2, 3	rankxpu 9800	. . . . 5 ⊢ (rank‘(𝐵 × 𝐴)) ⊆ suc suc (rank‘(𝐵 ∪ 𝐴))
10		uncom 4112	. . . . . . . 8 ⊢ (𝐵 ∪ 𝐴) = (𝐴 ∪ 𝐵)
11	10	fveq2i 6845	. . . . . . 7 ⊢ (rank‘(𝐵 ∪ 𝐴)) = (rank‘(𝐴 ∪ 𝐵))
12		suceq 6393	. . . . . . 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 6393	. . . . . 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 3984	. . . 4 ⊢ (rank‘(𝐵 × 𝐴)) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵))
17		rankon 9719	. . . . . 6 ⊢ (rank‘(𝐵 × 𝐴)) ∈ On
18	17	onordi 6438	. . . . 5 ⊢ Ord (rank‘(𝐵 × 𝐴))
19		rankon 9719	. . . . . . . 8 ⊢ (rank‘(𝐴 ∪ 𝐵)) ∈ On
20	19	onsuci 7791	. . . . . . 7 ⊢ suc (rank‘(𝐴 ∪ 𝐵)) ∈ On
21	20	onsuci 7791	. . . . . 6 ⊢ suc suc (rank‘(𝐴 ∪ 𝐵)) ∈ On
22	21	onordi 6438	. . . . 5 ⊢ Ord suc suc (rank‘(𝐴 ∪ 𝐵))
23		ordsucsssuc 7775	. . . . 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 693	. . . 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 3982	. 2 ⊢ (rank‘𝒫 (𝐵 × 𝐴)) ⊆ suc suc suc (rank‘(𝐴 ∪ 𝐵))
27	7, 26	sstri 3945	1 ⊢ (rank‘(𝐴 ↑m 𝐵)) ⊆ suc suc suc (rank‘(𝐴 ∪ 𝐵))

Syntax hints:   ↔ wb 206   = wceq 1542   ∈ wcel 2114  Vcvv 3442   ∪ cun 3901   ⊆ wss 3903  𝒫 cpw 4556   × cxp 5630  Ord word 6324  suc csuc 6327  ‘cfv 6500  (class class class)co 7368   ↑_m cmap 8775  rankcrnk 9687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-reg 9509  ax-inf2 9562

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-map 8777  df-pm 8778  df-r1 9688  df-rank 9689

This theorem is referenced by: (None)