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Mirrors > Home > ILE Home > Th. List > relcnvfi | GIF version |
Description: If a relation is finite, its converse is as well. (Contributed by Jim Kingdon, 5-Feb-2022.) |
Ref | Expression |
---|---|
relcnvfi | ⊢ ((Rel 𝐴 ∧ 𝐴 ∈ Fin) → ◡𝐴 ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrel2 4959 | . . . . 5 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴) | |
2 | 1 | biimpi 119 | . . . 4 ⊢ (Rel 𝐴 → ◡◡𝐴 = 𝐴) |
3 | 2 | adantr 274 | . . 3 ⊢ ((Rel 𝐴 ∧ 𝐴 ∈ Fin) → ◡◡𝐴 = 𝐴) |
4 | simpr 109 | . . 3 ⊢ ((Rel 𝐴 ∧ 𝐴 ∈ Fin) → 𝐴 ∈ Fin) | |
5 | 3, 4 | eqeltrd 2194 | . 2 ⊢ ((Rel 𝐴 ∧ 𝐴 ∈ Fin) → ◡◡𝐴 ∈ Fin) |
6 | relcnv 4887 | . . . 4 ⊢ Rel ◡𝐴 | |
7 | cnvexg 5046 | . . . 4 ⊢ (𝐴 ∈ Fin → ◡𝐴 ∈ V) | |
8 | cnven 6670 | . . . 4 ⊢ ((Rel ◡𝐴 ∧ ◡𝐴 ∈ V) → ◡𝐴 ≈ ◡◡𝐴) | |
9 | 6, 7, 8 | sylancr 410 | . . 3 ⊢ (𝐴 ∈ Fin → ◡𝐴 ≈ ◡◡𝐴) |
10 | 9 | adantl 275 | . 2 ⊢ ((Rel 𝐴 ∧ 𝐴 ∈ Fin) → ◡𝐴 ≈ ◡◡𝐴) |
11 | enfii 6736 | . 2 ⊢ ((◡◡𝐴 ∈ Fin ∧ ◡𝐴 ≈ ◡◡𝐴) → ◡𝐴 ∈ Fin) | |
12 | 5, 10, 11 | syl2anc 408 | 1 ⊢ ((Rel 𝐴 ∧ 𝐴 ∈ Fin) → ◡𝐴 ∈ Fin) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1316 ∈ wcel 1465 Vcvv 2660 class class class wbr 3899 ◡ccnv 4508 Rel wrel 4514 ≈ cen 6600 Fincfn 6602 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-1st 6006 df-2nd 6007 df-er 6397 df-en 6603 df-fin 6605 |
This theorem is referenced by: funrnfi 6798 fsumcnv 11174 |
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