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Mirrors > Home > MPE Home > Th. List > f1fi | Structured version Visualization version GIF version |
Description: If a 1-to-1 function has a finite codomain its domain is finite. (Contributed by FL, 31-Jul-2009.) (Revised by Mario Carneiro, 24-Jun-2015.) |
Ref | Expression |
---|---|
f1fi | ⊢ ((𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1f 6014 | . . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵) | |
2 | frn 5966 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → ran 𝐹 ⊆ 𝐵) |
4 | ssfi 8065 | . . 3 ⊢ ((𝐵 ∈ Fin ∧ ran 𝐹 ⊆ 𝐵) → ran 𝐹 ∈ Fin) | |
5 | 3, 4 | sylan2 490 | . 2 ⊢ ((𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → ran 𝐹 ∈ Fin) |
6 | f1f1orn 6061 | . . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→ran 𝐹) | |
7 | 6 | adantl 481 | . . 3 ⊢ ((𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹:𝐴–1-1-onto→ran 𝐹) |
8 | f1ocnv 6062 | . . 3 ⊢ (𝐹:𝐴–1-1-onto→ran 𝐹 → ◡𝐹:ran 𝐹–1-1-onto→𝐴) | |
9 | f1ofo 6057 | . . 3 ⊢ (◡𝐹:ran 𝐹–1-1-onto→𝐴 → ◡𝐹:ran 𝐹–onto→𝐴) | |
10 | 7, 8, 9 | 3syl 18 | . 2 ⊢ ((𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → ◡𝐹:ran 𝐹–onto→𝐴) |
11 | fofi 8135 | . 2 ⊢ ((ran 𝐹 ∈ Fin ∧ ◡𝐹:ran 𝐹–onto→𝐴) → 𝐴 ∈ Fin) | |
12 | 5, 10, 11 | syl2anc 691 | 1 ⊢ ((𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 ⊆ wss 3540 ◡ccnv 5037 ran crn 5039 ⟶wf 5800 –1-1→wf1 5801 –onto→wfo 5802 –1-1-onto→wf1o 5803 Fincfn 7841 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-om 6958 df-1o 7447 df-er 7629 df-en 7842 df-dom 7843 df-fin 7845 |
This theorem is referenced by: ixpfi2 8147 fsumvma 24738 edgusgranbfin 25979 fourierdlem51 39050 prminf2 40038 edgusgrnbfin 40601 |
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