Theorem konigth 10463

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 6822	. . . . 5 ⊢ (𝑏 = 𝑎 → (𝑓‘𝑏) = (𝑓‘𝑎))
5	4	fveq1d 6824	. . . 4 ⊢ (𝑏 = 𝑎 → ((𝑓‘𝑏)‘𝑖) = ((𝑓‘𝑎)‘𝑖))
6	5	cbvmptv 5196	. . 3 ⊢ (𝑏 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑏)‘𝑖)) = (𝑎 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑎)‘𝑖))
7	6	mpteq2i 5188	. 2 ⊢ (𝑖 ∈ 𝐴 ↦ (𝑏 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑏)‘𝑖))) = (𝑖 ∈ 𝐴 ↦ (𝑎 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑎)‘𝑖)))
8		fveq2 6822	. . 3 ⊢ (𝑗 = 𝑖 → (𝑒‘𝑗) = (𝑒‘𝑖))
9	8	cbvmptv 5196	. 2 ⊢ (𝑗 ∈ 𝐴 ↦ (𝑒‘𝑗)) = (𝑖 ∈ 𝐴 ↦ (𝑒‘𝑖))
10	1, 2, 3, 7, 9	konigthlem 10462	1 ⊢ (∀𝑖 ∈ 𝐴 (𝑀‘𝑖) ≺ (𝑁‘𝑖) → 𝑆 ≺ 𝑃)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3436  ∪ ciun 4941   class class class wbr 5092   ↦ cmpt 5173  ‘cfv 6482  Xcixp 8824   ≺ csdm 8871

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-ac2 10357

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-er 8625  df-map 8755  df-ixp 8825  df-en 8873  df-dom 8874  df-sdom 8875  df-card 9835  df-acn 9838  df-ac 10010

This theorem is referenced by:  pwcfsdom  10477