Theorem erdszelem1 31299

Description: Lemma for erdsze 31310. (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 6718	. . . 4 ⊢ (1...𝐴) ∈ V
2	1	elpw2 4858	. . 3 ⊢ (𝑋 ∈ 𝒫 (1...𝐴) ↔ 𝑋 ⊆ (1...𝐴))
3	2	anbi1i 731	. 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 5423	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝐹 ↾ 𝑦) = (𝐹 ↾ 𝑋))
5		isoeq1 6607	. . . . . 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 6610	. . . . 5 ⊢ (𝑦 = 𝑋 → ((𝐹 ↾ 𝑋) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ↔ (𝐹 ↾ 𝑋) Isom < , 𝑂 (𝑋, (𝐹 “ 𝑦))))
8		imaeq2 5497	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝐹 “ 𝑦) = (𝐹 “ 𝑋))
9		isoeq5 6611	. . . . . 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 294	. . . 4 ⊢ (𝑦 = 𝑋 → ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ↔ (𝐹 ↾ 𝑋) Isom < , 𝑂 (𝑋, (𝐹 “ 𝑋))))
12		eleq2 2719	. . . 4 ⊢ (𝑦 = 𝑋 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑋))
13	11, 12	anbi12d 747	. . 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 3399	. 2 ⊢ (𝑋 ∈ 𝑆 ↔ (𝑋 ∈ 𝒫 (1...𝐴) ∧ ((𝐹 ↾ 𝑋) Isom < , 𝑂 (𝑋, (𝐹 “ 𝑋)) ∧ 𝐴 ∈ 𝑋)))
16		3anass 1059	. 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 292	1 ⊢ (𝑋 ∈ 𝑆 ↔ (𝑋 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑋) Isom < , 𝑂 (𝑋, (𝐹 “ 𝑋)) ∧ 𝐴 ∈ 𝑋))

Syntax hints:   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  {crab 2945   ⊆ wss 3607  𝒫 cpw 4191   ↾ cres 5145   “ cima 5146   Isom wiso 5927  (class class class)co 6690  1c1 9975   < clt 10112  ...cfz 12364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-ov 6693

This theorem is referenced by:  erdszelem2  31300  erdszelem4  31302  erdszelem7  31305  erdszelem8  31306