Theorem konigth 10581

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 6882	. . . . 5 ⊢ (𝑏 = 𝑎 → (𝑓‘𝑏) = (𝑓‘𝑎))
5	4	fveq1d 6884	. . . 4 ⊢ (𝑏 = 𝑎 → ((𝑓‘𝑏)‘𝑖) = ((𝑓‘𝑎)‘𝑖))
6	5	cbvmptv 5213	. . 3 ⊢ (𝑏 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑏)‘𝑖)) = (𝑎 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑎)‘𝑖))
7	6	mpteq2i 5205	. 2 ⊢ (𝑖 ∈ 𝐴 ↦ (𝑏 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑏)‘𝑖))) = (𝑖 ∈ 𝐴 ↦ (𝑎 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑎)‘𝑖)))
8		fveq2 6882	. . 3 ⊢ (𝑗 = 𝑖 → (𝑒‘𝑗) = (𝑒‘𝑖))
9	8	cbvmptv 5213	. 2 ⊢ (𝑗 ∈ 𝐴 ↦ (𝑒‘𝑗)) = (𝑖 ∈ 𝐴 ↦ (𝑒‘𝑖))
10	1, 2, 3, 7, 9	konigthlem 10580	1 ⊢ (∀𝑖 ∈ 𝐴 (𝑀‘𝑖) ≺ (𝑁‘𝑖) → 𝑆 ≺ 𝑃)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ∀wral 3078  Vcvv 3453  ∪ ciun 4954   class class class wbr 5107   ↦ cmpt 5190  ‘cfv 6537  Xcixp 8907   ≺ csdm 8954

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-ac2 10468

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-er 8699  df-map 8831  df-ixp 8908  df-en 8956  df-dom 8957  df-sdom 8958  df-card 9947  df-acn 9950  df-ac 10122

This theorem is used by:  pwcfsdom  10595