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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 6672 | . . . . 5 ⊢ (𝑏 = 𝑎 → (𝑓‘𝑏) = (𝑓‘𝑎)) | |
5 | 4 | fveq1d 6674 | . . . 4 ⊢ (𝑏 = 𝑎 → ((𝑓‘𝑏)‘𝑖) = ((𝑓‘𝑎)‘𝑖)) |
6 | 5 | cbvmptv 5171 | . . 3 ⊢ (𝑏 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑏)‘𝑖)) = (𝑎 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑎)‘𝑖)) |
7 | 6 | mpteq2i 5160 | . 2 ⊢ (𝑖 ∈ 𝐴 ↦ (𝑏 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑏)‘𝑖))) = (𝑖 ∈ 𝐴 ↦ (𝑎 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑎)‘𝑖))) |
8 | fveq2 6672 | . . 3 ⊢ (𝑗 = 𝑖 → (𝑒‘𝑗) = (𝑒‘𝑖)) | |
9 | 8 | cbvmptv 5171 | . 2 ⊢ (𝑗 ∈ 𝐴 ↦ (𝑒‘𝑗)) = (𝑖 ∈ 𝐴 ↦ (𝑒‘𝑖)) |
10 | 1, 2, 3, 7, 9 | konigthlem 9992 | 1 ⊢ (∀𝑖 ∈ 𝐴 (𝑀‘𝑖) ≺ (𝑁‘𝑖) → 𝑆 ≺ 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∀wral 3140 Vcvv 3496 ∪ ciun 4921 class class class wbr 5068 ↦ cmpt 5148 ‘cfv 6357 Xcixp 8463 ≺ csdm 8510 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-ac2 9887 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-er 8291 df-map 8410 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-card 9370 df-acn 9373 df-ac 9544 |
This theorem is referenced by: pwcfsdom 10007 |
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