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Theorem konigth 9270
 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 regular 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.)
Hypotheses
Ref Expression
konigth.1 𝐴 ∈ V
konigth.2 𝑆 = 𝑖𝐴 (𝑀𝑖)
konigth.3 𝑃 = X𝑖𝐴 (𝑁𝑖)
Assertion
Ref Expression
konigth (∀𝑖𝐴 (𝑀𝑖) ≺ (𝑁𝑖) → 𝑆𝑃)
Distinct variable group:   𝐴,𝑖
Allowed substitution hints:   𝑃(𝑖)   𝑆(𝑖)   𝑀(𝑖)   𝑁(𝑖)

Proof of Theorem konigth
Dummy variables 𝑎 𝑒 𝑓 𝑗 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 konigth.1 . 2 𝐴 ∈ V
2 konigth.2 . 2 𝑆 = 𝑖𝐴 (𝑀𝑖)
3 konigth.3 . 2 𝑃 = X𝑖𝐴 (𝑁𝑖)
4 fveq2 6103 . . . . 5 (𝑏 = 𝑎 → (𝑓𝑏) = (𝑓𝑎))
54fveq1d 6105 . . . 4 (𝑏 = 𝑎 → ((𝑓𝑏)‘𝑖) = ((𝑓𝑎)‘𝑖))
65cbvmptv 4678 . . 3 (𝑏 ∈ (𝑀𝑖) ↦ ((𝑓𝑏)‘𝑖)) = (𝑎 ∈ (𝑀𝑖) ↦ ((𝑓𝑎)‘𝑖))
76mpteq2i 4669 . 2 (𝑖𝐴 ↦ (𝑏 ∈ (𝑀𝑖) ↦ ((𝑓𝑏)‘𝑖))) = (𝑖𝐴 ↦ (𝑎 ∈ (𝑀𝑖) ↦ ((𝑓𝑎)‘𝑖)))
8 fveq2 6103 . . 3 (𝑗 = 𝑖 → (𝑒𝑗) = (𝑒𝑖))
98cbvmptv 4678 . 2 (𝑗𝐴 ↦ (𝑒𝑗)) = (𝑖𝐴 ↦ (𝑒𝑖))
101, 2, 3, 7, 9konigthlem 9269 1 (∀𝑖𝐴 (𝑀𝑖) ≺ (𝑁𝑖) → 𝑆𝑃)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  ∪ ciun 4455   class class class wbr 4583   ↦ cmpt 4643  ‘cfv 5804  Xcixp 7794   ≺ csdm 7840 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-ac2 9168 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-er 7629  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-card 8648  df-acn 8651  df-ac 8822 This theorem is referenced by:  pwcfsdom  9284
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