Theorem erdszelem1 36080

Description: Lemma for erdsze 36091. (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 7463	. . . 4 ⊢ (1...𝐴) ∈ V
2	1	elpw2 5300	. . 3 ⊢ (𝑋 ∈ 𝒫 (1...𝐴) ↔ 𝑋 ⊆ (1...𝐴))
3	2	anbi1i 636	. 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 7335	. . . . . 6 ⊢ ((𝐹 ↾ 𝑦) = (𝐹 ↾ 𝑋) → ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ↔ (𝐹 ↾ 𝑋) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦))))
6	4, 5	syl 18	. . . . 5 ⊢ (𝑦 = 𝑋 → ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ↔ (𝐹 ↾ 𝑋) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦))))
7		isoeq4 7338	. . . . 5 ⊢ (𝑦 = 𝑋 → ((𝐹 ↾ 𝑋) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ↔ (𝐹 ↾ 𝑋) Isom < , 𝑂 (𝑋, (𝐹 “ 𝑦))))
8		imaeq2 6054	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝐹 “ 𝑦) = (𝐹 “ 𝑋))
9		isoeq5 7339	. . . . . 6 ⊢ ((𝐹 “ 𝑦) = (𝐹 “ 𝑋) → ((𝐹 ↾ 𝑋) Isom < , 𝑂 (𝑋, (𝐹 “ 𝑦)) ↔ (𝐹 ↾ 𝑋) Isom < , 𝑂 (𝑋, (𝐹 “ 𝑋))))
10	8, 9	syl 18	. . . . 5 ⊢ (𝑦 = 𝑋 → ((𝐹 ↾ 𝑋) Isom < , 𝑂 (𝑋, (𝐹 “ 𝑦)) ↔ (𝐹 ↾ 𝑋) Isom < , 𝑂 (𝑋, (𝐹 “ 𝑋))))
11	6, 7, 10	3bitrd 308	. . . 4 ⊢ (𝑦 = 𝑋 → ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ↔ (𝐹 ↾ 𝑋) Isom < , 𝑂 (𝑋, (𝐹 “ 𝑋))))
12		eleq2 2853	. . . 4 ⊢ (𝑦 = 𝑋 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑋))
13	11, 12	anbi12d 644	. . 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 3652	. 2 ⊢ (𝑋 ∈ 𝑆 ↔ (𝑋 ∈ 𝒫 (1...𝐴) ∧ ((𝐹 ↾ 𝑋) Isom < , 𝑂 (𝑋, (𝐹 “ 𝑋)) ∧ 𝐴 ∈ 𝑋)))
16		3anass 1111	. 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 306	1 ⊢ (𝑋 ∈ 𝑆 ↔ (𝑋 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑋) Isom < , 𝑂 (𝑋, (𝐹 “ 𝑋)) ∧ 𝐴 ∈ 𝑋))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2146  {crab 3416   ⊆ wss 3900  𝒫 cpw 4560   ↾ cres 5657   “ cima 5658   Isom wiso 6548  (class class class)co 7430  1c1 11214   < clt 11356  ...cfz 13652

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5252  ax-nul 5263

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-isom 6556  df-ov 7433

This theorem is used by:  erdszelem2  36081  erdszelem4  36083  erdszelem7  36086  erdszelem8  36087