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Theorem erdszelem3 30429
 Description: Lemma for erdsze 30438. (Contributed by Mario Carneiro, 22-Jan-2015.)
Hypotheses
Ref Expression
erdsze.n (𝜑𝑁 ∈ ℕ)
erdsze.f (𝜑𝐹:(1...𝑁)–1-1→ℝ)
erdszelem.k 𝐾 = (𝑥 ∈ (1...𝑁) ↦ sup((# “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)}), ℝ, < ))
Assertion
Ref Expression
erdszelem3 (𝐴 ∈ (1...𝑁) → (𝐾𝐴) = sup((# “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹𝑦) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ∧ 𝐴𝑦)}), ℝ, < ))
Distinct variable groups:   𝑥,𝑦,𝐹   𝑥,𝐴,𝑦   𝑥,𝑂,𝑦   𝑥,𝑁,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝐾(𝑥,𝑦)

Proof of Theorem erdszelem3
StepHypRef Expression
1 oveq2 6557 . . . . . 6 (𝑥 = 𝐴 → (1...𝑥) = (1...𝐴))
21pweqd 4113 . . . . 5 (𝑥 = 𝐴 → 𝒫 (1...𝑥) = 𝒫 (1...𝐴))
3 eleq1 2676 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑦𝐴𝑦))
43anbi2d 736 . . . . 5 (𝑥 = 𝐴 → (((𝐹𝑦) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦) ↔ ((𝐹𝑦) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ∧ 𝐴𝑦)))
52, 4rabeqbidv 3168 . . . 4 (𝑥 = 𝐴 → {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)} = {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹𝑦) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ∧ 𝐴𝑦)})
65imaeq2d 5385 . . 3 (𝑥 = 𝐴 → (# “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)}) = (# “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹𝑦) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ∧ 𝐴𝑦)}))
76supeq1d 8235 . 2 (𝑥 = 𝐴 → sup((# “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)}), ℝ, < ) = sup((# “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹𝑦) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ∧ 𝐴𝑦)}), ℝ, < ))
8 erdszelem.k . 2 𝐾 = (𝑥 ∈ (1...𝑁) ↦ sup((# “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹𝑦) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ∧ 𝑥𝑦)}), ℝ, < ))
9 ltso 9997 . . 3 < Or ℝ
109supex 8252 . 2 sup((# “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹𝑦) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ∧ 𝐴𝑦)}), ℝ, < ) ∈ V
117, 8, 10fvmpt 6191 1 (𝐴 ∈ (1...𝑁) → (𝐾𝐴) = sup((# “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹𝑦) Isom < , 𝑂 (𝑦, (𝐹𝑦)) ∧ 𝐴𝑦)}), ℝ, < ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  𝒫 cpw 4108   ↦ cmpt 4643   ↾ cres 5040   “ cima 5041  –1-1→wf1 5801  ‘cfv 5804   Isom wiso 5805  (class class class)co 6549  supcsup 8229  ℝcr 9814  1c1 9816   < clt 9953  ℕcn 10897  ...cfz 12197  #chash 12979 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-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-resscn 9872  ax-pre-lttri 9889  ax-pre-lttrn 9890 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-nel 2783  df-ral 2901  df-rex 2902  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-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  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-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-pnf 9955  df-mnf 9956  df-ltxr 9958 This theorem is referenced by:  erdszelem5  30431  erdszelem8  30434
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