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Mathbox for Jeff Hankins |
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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 6907 | . 2 ⊢ (𝑥 = 𝑦 → (topGen‘𝑥) = (topGen‘𝑦)) | |
2 | fneval.1 | . . . . . 6 ⊢ ∼ = (Fne ∩ ◡Fne) | |
3 | inss1 4245 | . . . . . 6 ⊢ (Fne ∩ ◡Fne) ⊆ Fne | |
4 | 2, 3 | eqsstri 4030 | . . . . 5 ⊢ ∼ ⊆ Fne |
5 | fnerel 36321 | . . . . 5 ⊢ Rel Fne | |
6 | relss 5794 | . . . . 5 ⊢ ( ∼ ⊆ Fne → (Rel Fne → Rel ∼ )) | |
7 | 4, 5, 6 | mp2 9 | . . . 4 ⊢ Rel ∼ |
8 | dfrel4v 6212 | . . . 4 ⊢ (Rel ∼ ↔ ∼ = {〈𝑥, 𝑦〉 ∣ 𝑥 ∼ 𝑦}) | |
9 | 7, 8 | mpbi 230 | . . 3 ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ 𝑥 ∼ 𝑦} |
10 | 2 | fneval 36335 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝑦 ↔ (topGen‘𝑥) = (topGen‘𝑦))) |
11 | 10 | el2v 3485 | . . . 4 ⊢ (𝑥 ∼ 𝑦 ↔ (topGen‘𝑥) = (topGen‘𝑦)) |
12 | 11 | opabbii 5215 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥 ∼ 𝑦} = {〈𝑥, 𝑦〉 ∣ (topGen‘𝑥) = (topGen‘𝑦)} |
13 | 9, 12 | eqtri 2763 | . 2 ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ (topGen‘𝑥) = (topGen‘𝑦)} |
14 | 1, 13 | eqer 8780 | 1 ⊢ ∼ Er V |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1537 Vcvv 3478 ∩ cin 3962 ⊆ wss 3963 class class class wbr 5148 {copab 5210 ◡ccnv 5688 Rel wrel 5694 ‘cfv 6563 Er wer 8741 topGenctg 17484 Fnecfne 36319 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-er 8744 df-topgen 17490 df-fne 36320 |
This theorem is referenced by: topfneec 36338 topfneec2 36339 |
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