Theorem konigth 10492

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.)

Hypotheses

Ref	Expression
konigth.1	⊢ 𝐴 ∈ V
konigth.2	⊢ 𝑆 = ∪ 𝑖 ∈ 𝐴 (𝑀‘𝑖)
konigth.3	⊢ 𝑃 = X𝑖 ∈ 𝐴 (𝑁‘𝑖)

Assertion

Ref	Expression
konigth	⊢ (∀𝑖 ∈ 𝐴 (𝑀‘𝑖) ≺ (𝑁‘𝑖) → 𝑆 ≺ 𝑃)

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.

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 6840	. . . . 5 ⊢ (𝑏 = 𝑎 → (𝑓‘𝑏) = (𝑓‘𝑎))
5	4	fveq1d 6842	. . . 4 ⊢ (𝑏 = 𝑎 → ((𝑓‘𝑏)‘𝑖) = ((𝑓‘𝑎)‘𝑖))
6	5	cbvmptv 5189	. . 3 ⊢ (𝑏 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑏)‘𝑖)) = (𝑎 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑎)‘𝑖))
7	6	mpteq2i 5181	. 2 ⊢ (𝑖 ∈ 𝐴 ↦ (𝑏 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑏)‘𝑖))) = (𝑖 ∈ 𝐴 ↦ (𝑎 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑎)‘𝑖)))
8		fveq2 6840	. . 3 ⊢ (𝑗 = 𝑖 → (𝑒‘𝑗) = (𝑒‘𝑖))
9	8	cbvmptv 5189	. 2 ⊢ (𝑗 ∈ 𝐴 ↦ (𝑒‘𝑗)) = (𝑖 ∈ 𝐴 ↦ (𝑒‘𝑖))
10	1, 2, 3, 7, 9	konigthlem 10491	1 ⊢ (∀𝑖 ∈ 𝐴 (𝑀‘𝑖) ≺ (𝑁‘𝑖) → 𝑆 ≺ 𝑃)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ∀wral 3051  Vcvv 3429  ∪ ciun 4933   class class class wbr 5085   ↦ cmpt 5166  ‘cfv 6498  Xcixp 8845   ≺ csdm 8892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-ac2 10385

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-er 8643  df-map 8775  df-ixp 8846  df-en 8894  df-dom 8895  df-sdom 8896  df-card 9863  df-acn 9866  df-ac 10038

This theorem is referenced by:  pwcfsdom  10506