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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 7814 | . . . . 5 ⊢ (𝑁 ∈ ω → Ord 𝑁) | |
2 | ordge1n0 8444 | . . . . 5 ⊢ (Ord 𝑁 → (1o ⊆ 𝑁 ↔ 𝑁 ≠ ∅)) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝑁 ∈ ω → (1o ⊆ 𝑁 ↔ 𝑁 ≠ ∅)) |
4 | 3 | biimprd 248 | . . 3 ⊢ (𝑁 ∈ ω → (𝑁 ≠ ∅ → 1o ⊆ 𝑁)) |
5 | 4 | imdistani 570 | . 2 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≠ ∅) → (𝑁 ∈ ω ∧ 1o ⊆ 𝑁)) |
6 | eqid 2733 | . . 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 35918 | . 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 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 Vcvv 3447 ⊆ wss 3914 ∅c0 4286 ifcif 4490 ⟨cop 4596 ∪ cuni 4869 × cxp 5635 Ord word 6320 suc csuc 6323 ‘cfv 6500 ∈ cmpo 7363 ωcom 7806 1st c1st 7923 1oc1o 8409 ↑↑cfinxp 35904 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-2o 8417 df-oadd 8420 df-en 8890 df-fin 8893 df-finxp 35905 |
This theorem is referenced by: finxp2o 35920 finxp3o 35921 finxp00 35923 |
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