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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 5059 | . . . . 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 2247 | . 2 ⊢ ((Rel 𝐴 ∧ 𝐴 ∈ Fin) → ◡◡𝐴 ∈ Fin) |
6 | relcnv 4987 | . . . 4 ⊢ Rel ◡𝐴 | |
7 | cnvexg 5146 | . . . 4 ⊢ (𝐴 ∈ Fin → ◡𝐴 ∈ V) | |
8 | cnven 6784 | . . . 4 ⊢ ((Rel ◡𝐴 ∧ ◡𝐴 ∈ V) → ◡𝐴 ≈ ◡◡𝐴) | |
9 | 6, 7, 8 | sylancr 412 | . . 3 ⊢ (𝐴 ∈ Fin → ◡𝐴 ≈ ◡◡𝐴) |
10 | 9 | adantl 275 | . 2 ⊢ ((Rel 𝐴 ∧ 𝐴 ∈ Fin) → ◡𝐴 ≈ ◡◡𝐴) |
11 | enfii 6850 | . 2 ⊢ ((◡◡𝐴 ∈ Fin ∧ ◡𝐴 ≈ ◡◡𝐴) → ◡𝐴 ∈ Fin) | |
12 | 5, 10, 11 | syl2anc 409 | 1 ⊢ ((Rel 𝐴 ∧ 𝐴 ∈ Fin) → ◡𝐴 ∈ Fin) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1348 ∈ wcel 2141 Vcvv 2730 class class class wbr 3987 ◡ccnv 4608 Rel wrel 4614 ≈ cen 6714 Fincfn 6716 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-1st 6117 df-2nd 6118 df-er 6511 df-en 6717 df-fin 6719 |
This theorem is referenced by: funrnfi 6917 fsumcnv 11393 fprodcnv 11581 |
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