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Mirrors > Home > MPE Home > Th. List > Mathboxes > filnetlem1 | Structured version Visualization version GIF version |
Description: Lemma for filnet 33627. 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 6663 | . . . 4 ⊢ (𝑥 = 𝐴 → (1st ‘𝑥) = (1st ‘𝐴)) | |
2 | 1 | sseq2d 3996 | . . 3 ⊢ (𝑥 = 𝐴 → ((1st ‘𝑦) ⊆ (1st ‘𝑥) ↔ (1st ‘𝑦) ⊆ (1st ‘𝐴))) |
3 | fveq2 6663 | . . . 4 ⊢ (𝑦 = 𝐵 → (1st ‘𝑦) = (1st ‘𝐵)) | |
4 | 3 | sseq1d 3995 | . . 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 5637 | 1 ⊢ (𝐴𝐷𝐵 ↔ ((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) ∧ (1st ‘𝐵) ⊆ (1st ‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ⊆ wss 3933 {csn 4557 ∪ ciun 4910 class class class wbr 5057 {copab 5119 × cxp 5546 ‘cfv 6348 1st c1st 7676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-xp 5554 df-iota 6307 df-fv 6356 |
This theorem is referenced by: filnetlem2 33624 filnetlem3 33625 filnetlem4 33626 |
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