Theorem erdszelem1 34710

Description: Lemma for erdsze 34721. (Contributed by Mario Carneiro, 22-Jan-2015.)

Hypothesis

Ref	Expression
erdszelem1.1	⊢ 𝑆 = {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}

Assertion

Ref	Expression
erdszelem1	⊢ (𝑋 ∈ 𝑆 ↔ (𝑋 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑋) Isom < , 𝑂 (𝑋, (𝐹 “ 𝑋)) ∧ 𝐴 ∈ 𝑋))

Distinct variable groups:   𝑦,𝐴   𝑦,𝐹   𝑦,𝑂   𝑦,𝑋

Allowed substitution hint:   𝑆(𝑦)

Proof of Theorem erdszelem1

Step	Hyp	Ref	Expression
1		ovex 7438	. . . 4 ⊢ (1...𝐴) ∈ V
2	1	elpw2 5338	. . 3 ⊢ (𝑋 ∈ 𝒫 (1...𝐴) ↔ 𝑋 ⊆ (1...𝐴))
3	2	anbi1i 623	. 2 ⊢ ((𝑋 ∈ 𝒫 (1...𝐴) ∧ ((𝐹 ↾ 𝑋) Isom < , 𝑂 (𝑋, (𝐹 “ 𝑋)) ∧ 𝐴 ∈ 𝑋)) ↔ (𝑋 ⊆ (1...𝐴) ∧ ((𝐹 ↾ 𝑋) Isom < , 𝑂 (𝑋, (𝐹 “ 𝑋)) ∧ 𝐴 ∈ 𝑋)))
4		reseq2 5970	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝐹 ↾ 𝑦) = (𝐹 ↾ 𝑋))
5		isoeq1 7310	. . . . . 6 ⊢ ((𝐹 ↾ 𝑦) = (𝐹 ↾ 𝑋) → ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ↔ (𝐹 ↾ 𝑋) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦))))
6	4, 5	syl 17	. . . . 5 ⊢ (𝑦 = 𝑋 → ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ↔ (𝐹 ↾ 𝑋) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦))))
7		isoeq4 7313	. . . . 5 ⊢ (𝑦 = 𝑋 → ((𝐹 ↾ 𝑋) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ↔ (𝐹 ↾ 𝑋) Isom < , 𝑂 (𝑋, (𝐹 “ 𝑦))))
8		imaeq2 6049	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝐹 “ 𝑦) = (𝐹 “ 𝑋))
9		isoeq5 7314	. . . . . 6 ⊢ ((𝐹 “ 𝑦) = (𝐹 “ 𝑋) → ((𝐹 ↾ 𝑋) Isom < , 𝑂 (𝑋, (𝐹 “ 𝑦)) ↔ (𝐹 ↾ 𝑋) Isom < , 𝑂 (𝑋, (𝐹 “ 𝑋))))
10	8, 9	syl 17	. . . . 5 ⊢ (𝑦 = 𝑋 → ((𝐹 ↾ 𝑋) Isom < , 𝑂 (𝑋, (𝐹 “ 𝑦)) ↔ (𝐹 ↾ 𝑋) Isom < , 𝑂 (𝑋, (𝐹 “ 𝑋))))
11	6, 7, 10	3bitrd 305	. . . 4 ⊢ (𝑦 = 𝑋 → ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ↔ (𝐹 ↾ 𝑋) Isom < , 𝑂 (𝑋, (𝐹 “ 𝑋))))
12		eleq2 2816	. . . 4 ⊢ (𝑦 = 𝑋 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑋))
13	11, 12	anbi12d 630	. . 3 ⊢ (𝑦 = 𝑋 → (((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦) ↔ ((𝐹 ↾ 𝑋) Isom < , 𝑂 (𝑋, (𝐹 “ 𝑋)) ∧ 𝐴 ∈ 𝑋)))
14		erdszelem1.1	. . 3 ⊢ 𝑆 = {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}
15	13, 14	elrab2 3681	. 2 ⊢ (𝑋 ∈ 𝑆 ↔ (𝑋 ∈ 𝒫 (1...𝐴) ∧ ((𝐹 ↾ 𝑋) Isom < , 𝑂 (𝑋, (𝐹 “ 𝑋)) ∧ 𝐴 ∈ 𝑋)))
16		3anass 1092	. 2 ⊢ ((𝑋 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑋) Isom < , 𝑂 (𝑋, (𝐹 “ 𝑋)) ∧ 𝐴 ∈ 𝑋) ↔ (𝑋 ⊆ (1...𝐴) ∧ ((𝐹 ↾ 𝑋) Isom < , 𝑂 (𝑋, (𝐹 “ 𝑋)) ∧ 𝐴 ∈ 𝑋)))
17	3, 15, 16	3bitr4i 303	1 ⊢ (𝑋 ∈ 𝑆 ↔ (𝑋 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑋) Isom < , 𝑂 (𝑋, (𝐹 “ 𝑋)) ∧ 𝐴 ∈ 𝑋))

Syntax hints:   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {crab 3426   ⊆ wss 3943  𝒫 cpw 4597   ↾ cres 5671   “ cima 5672   Isom wiso 6538  (class class class)co 7405  1c1 11113   < clt 11252  ...cfz 13490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  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-ov 7408

This theorem is referenced by:  erdszelem2  34711  erdszelem4  34713  erdszelem7  34716  erdszelem8  34717