Theorem erdszelem3 35618

Description: Lemma for erdsze 35627. (Contributed by Mario Carneiro, 22-Jan-2015.)

Hypotheses

Ref	Expression
erdsze.n	⊢ (𝜑 → 𝑁 ∈ ℕ)
erdsze.f	⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ)
erdszelem.k	⊢ 𝐾 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ))

Assertion

Ref	Expression
erdszelem3	⊢ (𝐴 ∈ (1...𝑁) → (𝐾‘𝐴) = sup((♯ “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}), 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

Step	Hyp	Ref	Expression
1		oveq2 7419	. . . . . 6 ⊢ (𝑥 = 𝐴 → (1...𝑥) = (1...𝐴))
2	1	pweqd 4584	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝒫 (1...𝑥) = 𝒫 (1...𝐴))
3		eleq1 2857	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦))
4	3	anbi2d 641	. . . . 5 ⊢ (𝑥 = 𝐴 → (((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦) ↔ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)))
5	2, 4	rabeqbidv 3441	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)} = {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)})
6	5	imaeq2d 6063	. . 3 ⊢ (𝑥 = 𝐴 → (♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}) = (♯ “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}))
7	6	supeq1d 9406	. 2 ⊢ (𝑥 = 𝐴 → sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ) = sup((♯ “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}), ℝ, < ))
8		erdszelem.k	. 2 ⊢ 𝐾 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ))
9		ltso 11290	. . 3 ⊢ < Or ℝ
10	9	supex 9424	. 2 ⊢ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}), ℝ, < ) ∈ V
11	7, 8, 10	fvmpt 6990	1 ⊢ (𝐴 ∈ (1...𝑁) → (𝐾‘𝐴) = sup((♯ “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}), ℝ, < ))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  {crab 3423  𝒫 cpw 4567   ↦ cmpt 5196   ↾ cres 5664   “ cima 5665  –1-1→wf1 6534  ‘cfv 6537   Isom wiso 6538  (class class class)co 7411  supcsup 9400  ℝcr 11099  1c1 11101   < clt 11243  ℕcn 12233  ...cfz 13535  ♯chash 14366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-resscn 11157  ax-pre-lttri 11174  ax-pre-lttrn 11175

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-po 5570  df-so 5571  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  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-ov 7414  df-er 8694  df-en 8944  df-dom 8945  df-sdom 8946  df-sup 9402  df-pnf 11245  df-mnf 11246  df-ltxr 11248

This theorem is referenced by:  erdszelem5  35620  erdszelem8  35623