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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 6670 | . 2 ⊢ (𝑥 = 𝑦 → (topGen‘𝑥) = (topGen‘𝑦)) | |
2 | fneval.1 | . . . . . 6 ⊢ ∼ = (Fne ∩ ◡Fne) | |
3 | inss1 4205 | . . . . . 6 ⊢ (Fne ∩ ◡Fne) ⊆ Fne | |
4 | 2, 3 | eqsstri 4001 | . . . . 5 ⊢ ∼ ⊆ Fne |
5 | fnerel 33686 | . . . . 5 ⊢ Rel Fne | |
6 | relss 5656 | . . . . 5 ⊢ ( ∼ ⊆ Fne → (Rel Fne → Rel ∼ )) | |
7 | 4, 5, 6 | mp2 9 | . . . 4 ⊢ Rel ∼ |
8 | dfrel4v 6047 | . . . 4 ⊢ (Rel ∼ ↔ ∼ = {〈𝑥, 𝑦〉 ∣ 𝑥 ∼ 𝑦}) | |
9 | 7, 8 | mpbi 232 | . . 3 ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ 𝑥 ∼ 𝑦} |
10 | 2 | fneval 33700 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝑦 ↔ (topGen‘𝑥) = (topGen‘𝑦))) |
11 | 10 | el2v 3501 | . . . 4 ⊢ (𝑥 ∼ 𝑦 ↔ (topGen‘𝑥) = (topGen‘𝑦)) |
12 | 11 | opabbii 5133 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥 ∼ 𝑦} = {〈𝑥, 𝑦〉 ∣ (topGen‘𝑥) = (topGen‘𝑦)} |
13 | 9, 12 | eqtri 2844 | . 2 ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ (topGen‘𝑥) = (topGen‘𝑦)} |
14 | 1, 13 | eqer 8324 | 1 ⊢ ∼ Er V |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 Vcvv 3494 ∩ cin 3935 ⊆ wss 3936 class class class wbr 5066 {copab 5128 ◡ccnv 5554 Rel wrel 5560 ‘cfv 6355 Er wer 8286 topGenctg 16711 Fnecfne 33684 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-er 8289 df-topgen 16717 df-fne 33685 |
This theorem is referenced by: topfneec 33703 topfneec2 33704 |
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