filnetlem1 - Mathbox for Jeff Hankins

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Theorem filnetlem1 36366

Description: Lemma for filnet 36370. 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 6858	. . . 4 ⊢ (𝑥 = 𝐴 → (1^st ‘𝑥) = (1^st ‘𝐴))
2	1	sseq2d 3979	. . 3 ⊢ (𝑥 = 𝐴 → ((1^st ‘𝑦) ⊆ (1^st ‘𝑥) ↔ (1^st ‘𝑦) ⊆ (1^st ‘𝐴)))
3		fveq2 6858	. . . 4 ⊢ (𝑦 = 𝐵 → (1^st ‘𝑦) = (1^st ‘𝐵))
4	3	sseq1d 3978	. . 3 ⊢ (𝑦 = 𝐵 → ((1^st ‘𝑦) ⊆ (1^st ‘𝐴) ↔ (1^st ‘𝐵) ⊆ (1^st ‘𝐴)))
5	2, 4	sylan9bb 509	. 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 5732	1 ⊢ (𝐴𝐷𝐵 ↔ ((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) ∧ (1^st ‘𝐵) ⊆ (1^st ‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 Vcvv 3447 ⊆ wss 3914 {csn 4589 ∪ ciun 4955 class class class wbr 5107 {copab 5169 × cxp 5636 ‘cfv 6511 1^st c1st 7966

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pr 5387

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2721 df-clel 2803 df-ral 3045 df-rex 3054 df-rab 3406 df-v 3449 df-dif 3917 df-un 3919 df-ss 3931 df-nul 4297 df-if 4489 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-br 5108 df-opab 5170 df-xp 5644 df-iota 6464 df-fv 6519

This theorem is referenced by: filnetlem2 36367 filnetlem3 36368 filnetlem4 36369

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