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Mirrors > Home > MPE Home > Th. List > cnvfi | Structured version Visualization version GIF version |
Description: If a set is finite, its converse is as well. (Contributed by Mario Carneiro, 28-Dec-2014.) |
Ref | Expression |
---|---|
cnvfi | ⊢ (𝐴 ∈ Fin → ◡𝐴 ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvcnvss 6051 | . . 3 ⊢ ◡◡𝐴 ⊆ 𝐴 | |
2 | ssfi 8738 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ ◡◡𝐴 ⊆ 𝐴) → ◡◡𝐴 ∈ Fin) | |
3 | 1, 2 | mpan2 689 | . 2 ⊢ (𝐴 ∈ Fin → ◡◡𝐴 ∈ Fin) |
4 | relcnv 5967 | . . 3 ⊢ Rel ◡𝐴 | |
5 | cnvexg 7629 | . . 3 ⊢ (𝐴 ∈ Fin → ◡𝐴 ∈ V) | |
6 | cnven 8585 | . . 3 ⊢ ((Rel ◡𝐴 ∧ ◡𝐴 ∈ V) → ◡𝐴 ≈ ◡◡𝐴) | |
7 | 4, 5, 6 | sylancr 589 | . 2 ⊢ (𝐴 ∈ Fin → ◡𝐴 ≈ ◡◡𝐴) |
8 | enfii 8735 | . 2 ⊢ ((◡◡𝐴 ∈ Fin ∧ ◡𝐴 ≈ ◡◡𝐴) → ◡𝐴 ∈ Fin) | |
9 | 3, 7, 8 | syl2anc 586 | 1 ⊢ (𝐴 ∈ Fin → ◡𝐴 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 Vcvv 3494 ⊆ wss 3936 class class class wbr 5066 ◡ccnv 5554 Rel wrel 5560 ≈ cen 8506 Fincfn 8509 |
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-3or 1084 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-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-om 7581 df-1st 7689 df-2nd 7690 df-er 8289 df-en 8510 df-fin 8513 |
This theorem is referenced by: rnfi 8807 fsumcnv 15128 fprodcnv 15337 gsumcom3 19098 gsummpt2co 30686 |
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