filnetlem1 - Mathbox for Jeff Hankins

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Theorem filnetlem1 36909

Description: Lemma for filnet 36913. 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 6881	. . . 4 ⊢ (𝑥 = 𝐴 → (1^st ‘𝑥) = (1^st ‘𝐴))
2	1	sseq2d 3969	. . 3 ⊢ (𝑥 = 𝐴 → ((1^st ‘𝑦) ⊆ (1^st ‘𝑥) ↔ (1^st ‘𝑦) ⊆ (1^st ‘𝐴)))
3		fveq2 6881	. . . 4 ⊢ (𝑦 = 𝐵 → (1^st ‘𝑦) = (1^st ‘𝐵))
4	3	sseq1d 3968	. . 3 ⊢ (𝑦 = 𝐵 → ((1^st ‘𝑦) ⊆ (1^st ‘𝐴) ↔ (1^st ‘𝐵) ⊆ (1^st ‘𝐴)))
5	2, 4	sylan9bb 518	. 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 5754	1 ⊢ (𝐴𝐷𝐵 ↔ ((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) ∧ (1^st ‘𝐵) ⊆ (1^st ‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 209 ∧ wa 400 = wceq 1570 ∈ wcel 2143 Vcvv 3455 ⊆ wss 3905 {csn 4589 ∪ ciun 4956 class class class wbr 5109 {copab 5173 × cxp 5659 ‘cfv 6536 1^st c1st 7980

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-ext 2735 ax-sep 5257 ax-pr 5404

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-sb 2097 df-clab 2742 df-cleq 2755 df-clel 2838 df-ral 3080 df-rex 3090 df-rab 3417 df-v 3457 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4488 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-br 5110 df-opab 5174 df-xp 5667 df-iota 6492 df-fv 6544

This theorem is referenced by: filnetlem2 36910 filnetlem3 36911 filnetlem4 36912

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