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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 6695 | . 2 ⊢ (𝑥 = 𝑦 → (topGen‘𝑥) = (topGen‘𝑦)) | |
2 | fneval.1 | . . . . . 6 ⊢ ∼ = (Fne ∩ ◡Fne) | |
3 | inss1 4129 | . . . . . 6 ⊢ (Fne ∩ ◡Fne) ⊆ Fne | |
4 | 2, 3 | eqsstri 3921 | . . . . 5 ⊢ ∼ ⊆ Fne |
5 | fnerel 34213 | . . . . 5 ⊢ Rel Fne | |
6 | relss 5638 | . . . . 5 ⊢ ( ∼ ⊆ Fne → (Rel Fne → Rel ∼ )) | |
7 | 4, 5, 6 | mp2 9 | . . . 4 ⊢ Rel ∼ |
8 | dfrel4v 6033 | . . . 4 ⊢ (Rel ∼ ↔ ∼ = {〈𝑥, 𝑦〉 ∣ 𝑥 ∼ 𝑦}) | |
9 | 7, 8 | mpbi 233 | . . 3 ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ 𝑥 ∼ 𝑦} |
10 | 2 | fneval 34227 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝑦 ↔ (topGen‘𝑥) = (topGen‘𝑦))) |
11 | 10 | el2v 3406 | . . . 4 ⊢ (𝑥 ∼ 𝑦 ↔ (topGen‘𝑥) = (topGen‘𝑦)) |
12 | 11 | opabbii 5106 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥 ∼ 𝑦} = {〈𝑥, 𝑦〉 ∣ (topGen‘𝑥) = (topGen‘𝑦)} |
13 | 9, 12 | eqtri 2759 | . 2 ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ (topGen‘𝑥) = (topGen‘𝑦)} |
14 | 1, 13 | eqer 8404 | 1 ⊢ ∼ Er V |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1543 Vcvv 3398 ∩ cin 3852 ⊆ wss 3853 class class class wbr 5039 {copab 5101 ◡ccnv 5535 Rel wrel 5541 ‘cfv 6358 Er wer 8366 topGenctg 16896 Fnecfne 34211 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-iota 6316 df-fun 6360 df-fv 6366 df-er 8369 df-topgen 16902 df-fne 34212 |
This theorem is referenced by: topfneec 34230 topfneec2 34231 |
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