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Mirrors > Home > MPE Home > Th. List > f1oenfirn | Structured version Visualization version GIF version |
Description: If the range of a one-to-one, onto function is finite, then the domain and range of the function are equinumerous. (Contributed by BTernaryTau, 9-Sep-2024.) |
Ref | Expression |
---|---|
f1oenfirn | ⊢ ((𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1ocnv 6845 | . . . . 5 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴) | |
2 | f1ofn 6834 | . . . . . 6 ⊢ (◡𝐹:𝐵–1-1-onto→𝐴 → ◡𝐹 Fn 𝐵) | |
3 | fnfi 9187 | . . . . . 6 ⊢ ((◡𝐹 Fn 𝐵 ∧ 𝐵 ∈ Fin) → ◡𝐹 ∈ Fin) | |
4 | 2, 3 | sylan 579 | . . . . 5 ⊢ ((◡𝐹:𝐵–1-1-onto→𝐴 ∧ 𝐵 ∈ Fin) → ◡𝐹 ∈ Fin) |
5 | 1, 4 | sylan 579 | . . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐵 ∈ Fin) → ◡𝐹 ∈ Fin) |
6 | 5 | ancoms 458 | . . 3 ⊢ ((𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1-onto→𝐵) → ◡𝐹 ∈ Fin) |
7 | cnvfi 9186 | . . . 4 ⊢ (◡𝐹 ∈ Fin → ◡◡𝐹 ∈ Fin) | |
8 | f1orel 6836 | . . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Rel 𝐹) | |
9 | dfrel2 6188 | . . . . . . 7 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹) | |
10 | 8, 9 | sylib 217 | . . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡◡𝐹 = 𝐹) |
11 | 10 | eleq1d 2817 | . . . . 5 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (◡◡𝐹 ∈ Fin ↔ 𝐹 ∈ Fin)) |
12 | 11 | biimpac 478 | . . . 4 ⊢ ((◡◡𝐹 ∈ Fin ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐹 ∈ Fin) |
13 | 7, 12 | sylan 579 | . . 3 ⊢ ((◡𝐹 ∈ Fin ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐹 ∈ Fin) |
14 | 6, 13 | sylancom 587 | . 2 ⊢ ((𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐹 ∈ Fin) |
15 | f1oen3g 8968 | . 2 ⊢ ((𝐹 ∈ Fin ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵) | |
16 | 14, 15 | sylancom 587 | 1 ⊢ ((𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 class class class wbr 5148 ◡ccnv 5675 Rel wrel 5681 Fn wfn 6538 –1-1-onto→wf1o 6542 ≈ cen 8942 Fincfn 8945 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-om 7860 df-1o 8472 df-en 8946 df-fin 8949 |
This theorem is referenced by: ensymfib 9193 |
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