filnetlem1 - Mathbox for Jeff Hankins

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Theorem filnetlem1 33721

Description: Lemma for filnet 33725. Change variables. (Contributed by Jeff Hankins, 13-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)

**Hypotheses**
Ref	Expression
filnet.h	⊢ 𝐻 = ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛)
filnet.d	⊢ 𝐷 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ (1^st ‘𝑦) ⊆ (1^st ‘𝑥))}
filnetlem1.a	⊢ 𝐴 ∈ V
filnetlem1.b	⊢ 𝐵 ∈ V

**Assertion**
Ref	Expression
filnetlem1	⊢ (𝐴𝐷𝐵 ↔ ((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) ∧ (1^st ‘𝐵) ⊆ (1^st ‘𝐴)))

Distinct variable groups: 𝑥,𝑦,𝐴 𝑥,𝑛,𝑦,𝐹 𝑥,𝐻,𝑦 𝑥,𝐵,𝑦 Allowed substitution hints: 𝐴(𝑛) 𝐵(𝑛) 𝐷(𝑥,𝑦,𝑛) 𝐻(𝑛)

**Proof of Theorem filnetlem1**
Step	Hyp	Ref	Expression
1		fveq2 6665	. . . 4 ⊢ (𝑥 = 𝐴 → (1^st ‘𝑥) = (1^st ‘𝐴))
2	1	sseq2d 3999	. . 3 ⊢ (𝑥 = 𝐴 → ((1^st ‘𝑦) ⊆ (1^st ‘𝑥) ↔ (1^st ‘𝑦) ⊆ (1^st ‘𝐴)))
3		fveq2 6665	. . . 4 ⊢ (𝑦 = 𝐵 → (1^st ‘𝑦) = (1^st ‘𝐵))
4	3	sseq1d 3998	. . 3 ⊢ (𝑦 = 𝐵 → ((1^st ‘𝑦) ⊆ (1^st ‘𝐴) ↔ (1^st ‘𝐵) ⊆ (1^st ‘𝐴)))
5	2, 4	sylan9bb 512	. 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((1^st ‘𝑦) ⊆ (1^st ‘𝑥) ↔ (1^st ‘𝐵) ⊆ (1^st ‘𝐴)))
6		filnet.d	. 2 ⊢ 𝐷 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ (1^st ‘𝑦) ⊆ (1^st ‘𝑥))}
7	5, 6	brab2a 5639	1 ⊢ (𝐴𝐷𝐵 ↔ ((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) ∧ (1^st ‘𝐵) ⊆ (1^st ‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3495 ⊆ wss 3936 {csn 4561 ∪ ciun 4912 class class class wbr 5059 {copab 5121 × cxp 5548 ‘cfv 6350 1^st c1st 7681

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-xp 5556 df-iota 6309 df-fv 6358

This theorem is referenced by: filnetlem2 33722 filnetlem3 33723 filnetlem4 33724

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