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Mirrors > Home > MPE Home > Th. List > rankmapu | Structured version Visualization version GIF version |
Description: An upper bound on the rank of set exponentiation. (Contributed by Gérard Lang, 5-Aug-2018.) |
Ref | Expression |
---|---|
rankxpl.1 | ⊢ 𝐴 ∈ V |
rankxpl.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
rankmapu | ⊢ (rank‘(𝐴 ↑m 𝐵)) ⊆ suc suc suc (rank‘(𝐴 ∪ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapsspw 8916 | . . 3 ⊢ (𝐴 ↑m 𝐵) ⊆ 𝒫 (𝐵 × 𝐴) | |
2 | rankxpl.2 | . . . . . 6 ⊢ 𝐵 ∈ V | |
3 | rankxpl.1 | . . . . . 6 ⊢ 𝐴 ∈ V | |
4 | 2, 3 | xpex 7771 | . . . . 5 ⊢ (𝐵 × 𝐴) ∈ V |
5 | 4 | pwex 5385 | . . . 4 ⊢ 𝒫 (𝐵 × 𝐴) ∈ V |
6 | 5 | rankss 9886 | . . 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 9880 | . . 3 ⊢ (rank‘𝒫 (𝐵 × 𝐴)) = suc (rank‘(𝐵 × 𝐴)) |
9 | 2, 3 | rankxpu 9913 | . . . . 5 ⊢ (rank‘(𝐵 × 𝐴)) ⊆ suc suc (rank‘(𝐵 ∪ 𝐴)) |
10 | uncom 4167 | . . . . . . . 8 ⊢ (𝐵 ∪ 𝐴) = (𝐴 ∪ 𝐵) | |
11 | 10 | fveq2i 6909 | . . . . . . 7 ⊢ (rank‘(𝐵 ∪ 𝐴)) = (rank‘(𝐴 ∪ 𝐵)) |
12 | suceq 6451 | . . . . . . 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 6451 | . . . . . 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 4031 | . . . 4 ⊢ (rank‘(𝐵 × 𝐴)) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵)) |
17 | rankon 9832 | . . . . . 6 ⊢ (rank‘(𝐵 × 𝐴)) ∈ On | |
18 | 17 | onordi 6496 | . . . . 5 ⊢ Ord (rank‘(𝐵 × 𝐴)) |
19 | rankon 9832 | . . . . . . . 8 ⊢ (rank‘(𝐴 ∪ 𝐵)) ∈ On | |
20 | 19 | onsuci 7858 | . . . . . . 7 ⊢ suc (rank‘(𝐴 ∪ 𝐵)) ∈ On |
21 | 20 | onsuci 7858 | . . . . . 6 ⊢ suc suc (rank‘(𝐴 ∪ 𝐵)) ∈ On |
22 | 21 | onordi 6496 | . . . . 5 ⊢ Ord suc suc (rank‘(𝐴 ∪ 𝐵)) |
23 | ordsucsssuc 7842 | . . . . 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 4029 | . 2 ⊢ (rank‘𝒫 (𝐵 × 𝐴)) ⊆ suc suc suc (rank‘(𝐴 ∪ 𝐵)) |
27 | 7, 26 | sstri 4004 | 1 ⊢ (rank‘(𝐴 ↑m 𝐵)) ⊆ suc suc suc (rank‘(𝐴 ∪ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1536 ∈ wcel 2105 Vcvv 3477 ∪ cun 3960 ⊆ wss 3962 𝒫 cpw 4604 × cxp 5686 Ord word 6384 suc csuc 6387 ‘cfv 6562 (class class class)co 7430 ↑m cmap 8864 rankcrnk 9800 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-reg 9629 ax-inf2 9678 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-map 8866 df-pm 8867 df-r1 9801 df-rank 9802 |
This theorem is referenced by: (None) |
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