filnetlem1 - Mathbox for Jeff Hankins

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Theorem filnetlem1 36778

Description: Lemma for filnet 36782. 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 6882	. . . 4 ⊢ (𝑥 = 𝐴 → (1^st ‘𝑥) = (1^st ‘𝐴))
2	1	sseq2d 3977	. . 3 ⊢ (𝑥 = 𝐴 → ((1^st ‘𝑦) ⊆ (1^st ‘𝑥) ↔ (1^st ‘𝑦) ⊆ (1^st ‘𝐴)))
3		fveq2 6882	. . . 4 ⊢ (𝑦 = 𝐵 → (1^st ‘𝑦) = (1^st ‘𝐵))
4	3	sseq1d 3976	. . 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 5755	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 1567 ∈ wcel 2149 Vcvv 3463 ⊆ wss 3913 {csn 4594 ∪ ciun 4960 class class class wbr 5113 {copab 5177 × cxp 5660 ‘cfv 6537 1^st c1st 7984

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-ext 2741 ax-sep 5261 ax-pr 5405

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-sb 2098 df-clab 2748 df-cleq 2761 df-clel 2844 df-ral 3086 df-rex 3096 df-rab 3424 df-v 3465 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-nul 4295 df-if 4493 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-br 5114 df-opab 5178 df-xp 5668 df-iota 6493 df-fv 6545

This theorem is referenced by: filnetlem2 36779 filnetlem3 36780 filnetlem4 36781

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