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Mirrors > Home > MPE Home > Th. List > konigth | Structured version Visualization version GIF version |
Description: Konig's Theorem. If 𝑚(𝑖) ≺ 𝑛(𝑖) for all 𝑖 ∈ 𝐴, then Σ𝑖 ∈ 𝐴𝑚(𝑖) ≺ ∏𝑖 ∈ 𝐴𝑛(𝑖), where the sums and products stand in for disjoint union and infinite cartesian product. The version here is proven with unions rather than disjoint unions for convenience, but the version with disjoint unions is clearly a special case of this version. The Axiom of Choice is needed for this proof, but it contains AC as a simple corollary (letting 𝑚(𝑖) = ∅, this theorem says that an infinite cartesian product of nonempty sets is nonempty), so this is an AC equivalent. Theorem 11.26 of [TakeutiZaring] p. 107. (Contributed by Mario Carneiro, 22-Feb-2013.) |
Ref | Expression |
---|---|
konigth.1 | ⊢ 𝐴 ∈ V |
konigth.2 | ⊢ 𝑆 = ∪ 𝑖 ∈ 𝐴 (𝑀‘𝑖) |
konigth.3 | ⊢ 𝑃 = X𝑖 ∈ 𝐴 (𝑁‘𝑖) |
Ref | Expression |
---|---|
konigth | ⊢ (∀𝑖 ∈ 𝐴 (𝑀‘𝑖) ≺ (𝑁‘𝑖) → 𝑆 ≺ 𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | konigth.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | konigth.2 | . 2 ⊢ 𝑆 = ∪ 𝑖 ∈ 𝐴 (𝑀‘𝑖) | |
3 | konigth.3 | . 2 ⊢ 𝑃 = X𝑖 ∈ 𝐴 (𝑁‘𝑖) | |
4 | fveq2 6774 | . . . . 5 ⊢ (𝑏 = 𝑎 → (𝑓‘𝑏) = (𝑓‘𝑎)) | |
5 | 4 | fveq1d 6776 | . . . 4 ⊢ (𝑏 = 𝑎 → ((𝑓‘𝑏)‘𝑖) = ((𝑓‘𝑎)‘𝑖)) |
6 | 5 | cbvmptv 5187 | . . 3 ⊢ (𝑏 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑏)‘𝑖)) = (𝑎 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑎)‘𝑖)) |
7 | 6 | mpteq2i 5179 | . 2 ⊢ (𝑖 ∈ 𝐴 ↦ (𝑏 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑏)‘𝑖))) = (𝑖 ∈ 𝐴 ↦ (𝑎 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑎)‘𝑖))) |
8 | fveq2 6774 | . . 3 ⊢ (𝑗 = 𝑖 → (𝑒‘𝑗) = (𝑒‘𝑖)) | |
9 | 8 | cbvmptv 5187 | . 2 ⊢ (𝑗 ∈ 𝐴 ↦ (𝑒‘𝑗)) = (𝑖 ∈ 𝐴 ↦ (𝑒‘𝑖)) |
10 | 1, 2, 3, 7, 9 | konigthlem 10324 | 1 ⊢ (∀𝑖 ∈ 𝐴 (𝑀‘𝑖) ≺ (𝑁‘𝑖) → 𝑆 ≺ 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 ∀wral 3064 Vcvv 3432 ∪ ciun 4924 class class class wbr 5074 ↦ cmpt 5157 ‘cfv 6433 Xcixp 8685 ≺ csdm 8732 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-ac2 10219 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-er 8498 df-map 8617 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-card 9697 df-acn 9700 df-ac 9872 |
This theorem is referenced by: pwcfsdom 10339 |
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