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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 6906 | . 2 ⊢ (𝑥 = 𝑦 → (topGen‘𝑥) = (topGen‘𝑦)) | |
| 2 | fneval.1 | . . . . . 6 ⊢ ∼ = (Fne ∩ ◡Fne) | |
| 3 | inss1 4237 | . . . . . 6 ⊢ (Fne ∩ ◡Fne) ⊆ Fne | |
| 4 | 2, 3 | eqsstri 4030 | . . . . 5 ⊢ ∼ ⊆ Fne | 
| 5 | fnerel 36339 | . . . . 5 ⊢ Rel Fne | |
| 6 | relss 5791 | . . . . 5 ⊢ ( ∼ ⊆ Fne → (Rel Fne → Rel ∼ )) | |
| 7 | 4, 5, 6 | mp2 9 | . . . 4 ⊢ Rel ∼ | 
| 8 | dfrel4v 6210 | . . . 4 ⊢ (Rel ∼ ↔ ∼ = {〈𝑥, 𝑦〉 ∣ 𝑥 ∼ 𝑦}) | |
| 9 | 7, 8 | mpbi 230 | . . 3 ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ 𝑥 ∼ 𝑦} | 
| 10 | 2 | fneval 36353 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ∼ 𝑦 ↔ (topGen‘𝑥) = (topGen‘𝑦))) | 
| 11 | 10 | el2v 3487 | . . . 4 ⊢ (𝑥 ∼ 𝑦 ↔ (topGen‘𝑥) = (topGen‘𝑦)) | 
| 12 | 11 | opabbii 5210 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥 ∼ 𝑦} = {〈𝑥, 𝑦〉 ∣ (topGen‘𝑥) = (topGen‘𝑦)} | 
| 13 | 9, 12 | eqtri 2765 | . 2 ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ (topGen‘𝑥) = (topGen‘𝑦)} | 
| 14 | 1, 13 | eqer 8781 | 1 ⊢ ∼ Er V | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 = wceq 1540 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 class class class wbr 5143 {copab 5205 ◡ccnv 5684 Rel wrel 5690 ‘cfv 6561 Er wer 8742 topGenctg 17482 Fnecfne 36337 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-er 8745 df-topgen 17488 df-fne 36338 | 
| This theorem is referenced by: topfneec 36356 topfneec2 36357 | 
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