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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 6797 | . . . . 5 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴) | |
2 | f1ofn 6786 | . . . . . 6 ⊢ (◡𝐹:𝐵–1-1-onto→𝐴 → ◡𝐹 Fn 𝐵) | |
3 | fnfi 9128 | . . . . . 6 ⊢ ((◡𝐹 Fn 𝐵 ∧ 𝐵 ∈ Fin) → ◡𝐹 ∈ Fin) | |
4 | 2, 3 | sylan 581 | . . . . 5 ⊢ ((◡𝐹:𝐵–1-1-onto→𝐴 ∧ 𝐵 ∈ Fin) → ◡𝐹 ∈ Fin) |
5 | 1, 4 | sylan 581 | . . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐵 ∈ Fin) → ◡𝐹 ∈ Fin) |
6 | 5 | ancoms 460 | . . 3 ⊢ ((𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1-onto→𝐵) → ◡𝐹 ∈ Fin) |
7 | cnvfi 9127 | . . . 4 ⊢ (◡𝐹 ∈ Fin → ◡◡𝐹 ∈ Fin) | |
8 | f1orel 6788 | . . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Rel 𝐹) | |
9 | dfrel2 6142 | . . . . . . 7 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹) | |
10 | 8, 9 | sylib 217 | . . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡◡𝐹 = 𝐹) |
11 | 10 | eleq1d 2819 | . . . . 5 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (◡◡𝐹 ∈ Fin ↔ 𝐹 ∈ Fin)) |
12 | 11 | biimpac 480 | . . . 4 ⊢ ((◡◡𝐹 ∈ Fin ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐹 ∈ Fin) |
13 | 7, 12 | sylan 581 | . . 3 ⊢ ((◡𝐹 ∈ Fin ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐹 ∈ Fin) |
14 | 6, 13 | sylancom 589 | . 2 ⊢ ((𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐹 ∈ Fin) |
15 | f1oen3g 8909 | . 2 ⊢ ((𝐹 ∈ Fin ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵) | |
16 | 14, 15 | sylancom 589 | 1 ⊢ ((𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 class class class wbr 5106 ◡ccnv 5633 Rel wrel 5639 Fn wfn 6492 –1-1-onto→wf1o 6496 ≈ cen 8883 Fincfn 8886 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-om 7804 df-1o 8413 df-en 8887 df-fin 8890 |
This theorem is referenced by: ensymfib 9134 |
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