filnetlem1 - Mathbox for Jeff Hankins

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Theorem filnetlem1 36948

Description: Lemma for filnet 36952. 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 6885	. . . 4 ⊢ (𝑥 = 𝐴 → (1^st ‘𝑥) = (1^st ‘𝐴))
2	1	sseq2d 3970	. . 3 ⊢ (𝑥 = 𝐴 → ((1^st ‘𝑦) ⊆ (1^st ‘𝑥) ↔ (1^st ‘𝑦) ⊆ (1^st ‘𝐴)))
3		fveq2 6885	. . . 4 ⊢ (𝑦 = 𝐵 → (1^st ‘𝑦) = (1^st ‘𝐵))
4	3	sseq1d 3969	. . 3 ⊢ (𝑦 = 𝐵 → ((1^st ‘𝑦) ⊆ (1^st ‘𝐴) ↔ (1^st ‘𝐵) ⊆ (1^st ‘𝐴)))
5	2, 4	sylan9bb 519	. 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 5756	1 ⊢ (𝐴𝐷𝐵 ↔ ((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) ∧ (1^st ‘𝐵) ⊆ (1^st ‘𝐴)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2146 Vcvv 3457 ⊆ wss 3906 {csn 4591 ∪ ciun 4958 class class class wbr 5111 {copab 5175 × cxp 5661 ‘cfv 6540 1^st c1st 7990

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-ext 2737 ax-sep 5259 ax-pr 5406

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-sb 2100 df-clab 2744 df-cleq 2757 df-clel 2840 df-ral 3082 df-rex 3092 df-rab 3419 df-v 3459 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4287 df-if 4490 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-br 5112 df-opab 5176 df-xp 5669 df-iota 6496 df-fv 6548

This theorem is used by: filnetlem2 36949 filnetlem3 36950 filnetlem4 36951

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