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Mirrors > Home > MPE Home > Th. List > Mathboxes > finxpsuc | Structured version Visualization version GIF version |
Description: The value of Cartesian exponentiation at a successor. (Contributed by ML, 24-Oct-2020.) |
Ref | Expression |
---|---|
finxpsuc | ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≠ ∅) → (𝑈↑↑suc 𝑁) = ((𝑈↑↑𝑁) × 𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnord 7073 | . . . . 5 ⊢ (𝑁 ∈ ω → Ord 𝑁) | |
2 | ordge1n0 7578 | . . . . 5 ⊢ (Ord 𝑁 → (1𝑜 ⊆ 𝑁 ↔ 𝑁 ≠ ∅)) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝑁 ∈ ω → (1𝑜 ⊆ 𝑁 ↔ 𝑁 ≠ ∅)) |
4 | 3 | biimprd 238 | . . 3 ⊢ (𝑁 ∈ ω → (𝑁 ≠ ∅ → 1𝑜 ⊆ 𝑁)) |
5 | 4 | imdistani 726 | . 2 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≠ ∅) → (𝑁 ∈ ω ∧ 1𝑜 ⊆ 𝑁)) |
6 | eqid 2622 | . . 3 ⊢ (𝑦 ∈ ω, 𝑥 ∈ V ↦ if((𝑦 = 1𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑦, (1st ‘𝑥)〉, 〈𝑦, 𝑥〉))) = (𝑦 ∈ ω, 𝑥 ∈ V ↦ if((𝑦 = 1𝑜 ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑦, (1st ‘𝑥)〉, 〈𝑦, 𝑥〉))) | |
7 | 6 | finxpsuclem 33234 | . 2 ⊢ ((𝑁 ∈ ω ∧ 1𝑜 ⊆ 𝑁) → (𝑈↑↑suc 𝑁) = ((𝑈↑↑𝑁) × 𝑈)) |
8 | 5, 7 | syl 17 | 1 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≠ ∅) → (𝑈↑↑suc 𝑁) = ((𝑈↑↑𝑁) × 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 ⊆ wss 3574 ∅c0 3915 ifcif 4086 〈cop 4183 ∪ cuni 4436 × cxp 5112 Ord word 5722 suc csuc 5725 ‘cfv 5888 ↦ cmpt2 6652 ωcom 7065 1st c1st 7166 1𝑜c1o 7553 ↑↑cfinxp 33220 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-reg 8497 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-finxp 33221 |
This theorem is referenced by: finxp2o 33236 finxp3o 33237 finxp00 33239 |
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