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Mirrors > Home > MPE Home > Th. List > Mathboxes > fneer | Structured version Visualization version GIF version |
Description: Fineness intersected with its converse is an equivalence relation. (Contributed by Jeff Hankins, 6-Oct-2009.) (Revised by Mario Carneiro, 11-Sep-2015.) |
Ref | Expression |
---|---|
fneval.1 | ⊢ ∼ = (Fne ∩ ◡Fne) |
Ref | Expression |
---|---|
fneer | ⊢ ∼ Er V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6103 | . 2 ⊢ (𝑥 = 𝑦 → (topGen‘𝑥) = (topGen‘𝑦)) | |
2 | fneval.1 | . . . . . 6 ⊢ ∼ = (Fne ∩ ◡Fne) | |
3 | inss1 3795 | . . . . . 6 ⊢ (Fne ∩ ◡Fne) ⊆ Fne | |
4 | 2, 3 | eqsstri 3598 | . . . . 5 ⊢ ∼ ⊆ Fne |
5 | fnerel 31503 | . . . . 5 ⊢ Rel Fne | |
6 | relss 5129 | . . . . 5 ⊢ ( ∼ ⊆ Fne → (Rel Fne → Rel ∼ )) | |
7 | 4, 5, 6 | mp2 9 | . . . 4 ⊢ Rel ∼ |
8 | dfrel4v 5503 | . . . 4 ⊢ (Rel ∼ ↔ ∼ = {〈𝑥, 𝑦〉 ∣ 𝑥 ∼ 𝑦}) | |
9 | 7, 8 | mpbi 219 | . . 3 ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ 𝑥 ∼ 𝑦} |
10 | vex 3176 | . . . . 5 ⊢ 𝑥 ∈ V | |
11 | vex 3176 | . . . . 5 ⊢ 𝑦 ∈ V | |
12 | 2 | fneval 31517 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝑦 ↔ (topGen‘𝑥) = (topGen‘𝑦))) |
13 | 10, 11, 12 | mp2an 704 | . . . 4 ⊢ (𝑥 ∼ 𝑦 ↔ (topGen‘𝑥) = (topGen‘𝑦)) |
14 | 13 | opabbii 4649 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥 ∼ 𝑦} = {〈𝑥, 𝑦〉 ∣ (topGen‘𝑥) = (topGen‘𝑦)} |
15 | 9, 14 | eqtri 2632 | . 2 ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ (topGen‘𝑥) = (topGen‘𝑦)} |
16 | 1, 15 | eqer 7664 | 1 ⊢ ∼ Er V |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 class class class wbr 4583 {copab 4642 ◡ccnv 5037 Rel wrel 5043 ‘cfv 5804 Er wer 7626 topGenctg 15921 Fnecfne 31501 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-er 7629 df-topgen 15927 df-fne 31502 |
This theorem is referenced by: topfneec 31520 topfneec2 31521 |
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