Theorem erdszelem1 32440

Description: Lemma for erdsze 32451. (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 7191	. . . 4 ⊢ (1...𝐴) ∈ V
2	1	elpw2 5250	. . 3 ⊢ (𝑋 ∈ 𝒫 (1...𝐴) ↔ 𝑋 ⊆ (1...𝐴))
3	2	anbi1i 625	. 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 5850	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝐹 ↾ 𝑦) = (𝐹 ↾ 𝑋))
5		isoeq1 7072	. . . . . 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 7075	. . . . 5 ⊢ (𝑦 = 𝑋 → ((𝐹 ↾ 𝑋) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ↔ (𝐹 ↾ 𝑋) Isom < , 𝑂 (𝑋, (𝐹 “ 𝑦))))
8		imaeq2 5927	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝐹 “ 𝑦) = (𝐹 “ 𝑋))
9		isoeq5 7076	. . . . . 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 307	. . . 4 ⊢ (𝑦 = 𝑋 → ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ↔ (𝐹 ↾ 𝑋) Isom < , 𝑂 (𝑋, (𝐹 “ 𝑋))))
12		eleq2 2903	. . . 4 ⊢ (𝑦 = 𝑋 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑋))
13	11, 12	anbi12d 632	. . 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 3685	. 2 ⊢ (𝑋 ∈ 𝑆 ↔ (𝑋 ∈ 𝒫 (1...𝐴) ∧ ((𝐹 ↾ 𝑋) Isom < , 𝑂 (𝑋, (𝐹 “ 𝑋)) ∧ 𝐴 ∈ 𝑋)))
16		3anass 1091	. 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 305	1 ⊢ (𝑋 ∈ 𝑆 ↔ (𝑋 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑋) Isom < , 𝑂 (𝑋, (𝐹 “ 𝑋)) ∧ 𝐴 ∈ 𝑋))

Syntax hints:   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  {crab 3144   ⊆ wss 3938  𝒫 cpw 4541   ↾ cres 5559   “ cima 5560   Isom wiso 6358  (class class class)co 7158  1c1 10540   < clt 10677  ...cfz 12895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-isom 6366  df-ov 7161

This theorem is referenced by:  erdszelem2  32441  erdszelem4  32443  erdszelem7  32446  erdszelem8  32447