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Mirrors > Home > MPE Home > Th. List > Mathboxes > filnetlem1 | Structured version Visualization version GIF version |
Description: Lemma for filnet 34785. Change variables. (Contributed by Jeff Hankins, 13-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.) |
Ref | Expression |
---|---|
filnet.h | ⊢ 𝐻 = ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛) |
filnet.d | ⊢ 𝐷 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ (1st ‘𝑦) ⊆ (1st ‘𝑥))} |
filnetlem1.a | ⊢ 𝐴 ∈ V |
filnetlem1.b | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
filnetlem1 | ⊢ (𝐴𝐷𝐵 ↔ ((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) ∧ (1st ‘𝐵) ⊆ (1st ‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6839 | . . . 4 ⊢ (𝑥 = 𝐴 → (1st ‘𝑥) = (1st ‘𝐴)) | |
2 | 1 | sseq2d 3974 | . . 3 ⊢ (𝑥 = 𝐴 → ((1st ‘𝑦) ⊆ (1st ‘𝑥) ↔ (1st ‘𝑦) ⊆ (1st ‘𝐴))) |
3 | fveq2 6839 | . . . 4 ⊢ (𝑦 = 𝐵 → (1st ‘𝑦) = (1st ‘𝐵)) | |
4 | 3 | sseq1d 3973 | . . 3 ⊢ (𝑦 = 𝐵 → ((1st ‘𝑦) ⊆ (1st ‘𝐴) ↔ (1st ‘𝐵) ⊆ (1st ‘𝐴))) |
5 | 2, 4 | sylan9bb 510 | . 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((1st ‘𝑦) ⊆ (1st ‘𝑥) ↔ (1st ‘𝐵) ⊆ (1st ‘𝐴))) |
6 | filnet.d | . 2 ⊢ 𝐷 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ (1st ‘𝑦) ⊆ (1st ‘𝑥))} | |
7 | 5, 6 | brab2a 5723 | 1 ⊢ (𝐴𝐷𝐵 ↔ ((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) ∧ (1st ‘𝐵) ⊆ (1st ‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3443 ⊆ wss 3908 {csn 4584 ∪ ciun 4952 class class class wbr 5103 {copab 5165 × cxp 5629 ‘cfv 6493 1st c1st 7911 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-xp 5637 df-iota 6445 df-fv 6501 |
This theorem is referenced by: filnetlem2 34782 filnetlem3 34783 filnetlem4 34784 |
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