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Mirrors > Home > MPE Home > Th. List > Mathboxes > filnetlem1 | Structured version Visualization version GIF version |
Description: Lemma for filnet 32377. 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 6191 | . . . 4 ⊢ (𝑥 = 𝐴 → (1st ‘𝑥) = (1st ‘𝐴)) | |
2 | 1 | sseq2d 3633 | . . 3 ⊢ (𝑥 = 𝐴 → ((1st ‘𝑦) ⊆ (1st ‘𝑥) ↔ (1st ‘𝑦) ⊆ (1st ‘𝐴))) |
3 | fveq2 6191 | . . . 4 ⊢ (𝑦 = 𝐵 → (1st ‘𝑦) = (1st ‘𝐵)) | |
4 | 3 | sseq1d 3632 | . . 3 ⊢ (𝑦 = 𝐵 → ((1st ‘𝑦) ⊆ (1st ‘𝐴) ↔ (1st ‘𝐵) ⊆ (1st ‘𝐴))) |
5 | 2, 4 | sylan9bb 736 | . 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 5194 | 1 ⊢ (𝐴𝐷𝐵 ↔ ((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) ∧ (1st ‘𝐵) ⊆ (1st ‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ⊆ wss 3574 {csn 4177 ∪ ciun 4520 class class class wbr 4653 {copab 4712 × cxp 5112 ‘cfv 5888 1st c1st 7166 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-iota 5851 df-fv 5896 |
This theorem is referenced by: filnetlem2 32374 filnetlem3 32375 filnetlem4 32376 |
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