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Mirrors > Home > MPE Home > Th. List > f1domfi | Structured version Visualization version GIF version |
Description: If the codomain of a one-to-one function is finite, then the function's domain is dominated by its codomain. This theorem is proved without using the Axiom of Replacement or the Axiom of Power Sets (unlike f1domg 8968). (Contributed by BTernaryTau, 25-Sep-2024.) |
Ref | Expression |
---|---|
f1domfi | ⊢ ((𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1cnv 6858 | . . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → ◡𝐹:ran 𝐹–1-1-onto→𝐴) | |
2 | f1f 6788 | . . . . . 6 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵) | |
3 | 2 | frnd 6726 | . . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → ran 𝐹 ⊆ 𝐵) |
4 | ssfi 9173 | . . . . 5 ⊢ ((𝐵 ∈ Fin ∧ ran 𝐹 ⊆ 𝐵) → ran 𝐹 ∈ Fin) | |
5 | 3, 4 | sylan2 594 | . . . 4 ⊢ ((𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → ran 𝐹 ∈ Fin) |
6 | f1ofn 6835 | . . . . 5 ⊢ (◡𝐹:ran 𝐹–1-1-onto→𝐴 → ◡𝐹 Fn ran 𝐹) | |
7 | fnfi 9181 | . . . . 5 ⊢ ((◡𝐹 Fn ran 𝐹 ∧ ran 𝐹 ∈ Fin) → ◡𝐹 ∈ Fin) | |
8 | 6, 7 | sylan 581 | . . . 4 ⊢ ((◡𝐹:ran 𝐹–1-1-onto→𝐴 ∧ ran 𝐹 ∈ Fin) → ◡𝐹 ∈ Fin) |
9 | 1, 5, 8 | syl2an2 685 | . . 3 ⊢ ((𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → ◡𝐹 ∈ Fin) |
10 | cnvfi 9180 | . . . 4 ⊢ (◡𝐹 ∈ Fin → ◡◡𝐹 ∈ Fin) | |
11 | f1rel 6791 | . . . . . . 7 ⊢ (𝐹:𝐴–1-1→𝐵 → Rel 𝐹) | |
12 | dfrel2 6189 | . . . . . . 7 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹) | |
13 | 11, 12 | sylib 217 | . . . . . 6 ⊢ (𝐹:𝐴–1-1→𝐵 → ◡◡𝐹 = 𝐹) |
14 | 13 | eleq1d 2819 | . . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → (◡◡𝐹 ∈ Fin ↔ 𝐹 ∈ Fin)) |
15 | 14 | biimpac 480 | . . . 4 ⊢ ((◡◡𝐹 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹 ∈ Fin) |
16 | 10, 15 | sylan 581 | . . 3 ⊢ ((◡𝐹 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹 ∈ Fin) |
17 | 9, 16 | sylancom 589 | . 2 ⊢ ((𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹 ∈ Fin) |
18 | f1dom3g 8963 | . . 3 ⊢ ((𝐹 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵) | |
19 | 18 | 3expib 1123 | . 2 ⊢ (𝐹 ∈ Fin → ((𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵)) |
20 | 17, 19 | mpcom 38 | 1 ⊢ ((𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ⊆ wss 3949 class class class wbr 5149 ◡ccnv 5676 ran crn 5678 Rel wrel 5682 Fn wfn 6539 –1-1→wf1 6541 –1-1-onto→wf1o 6543 ≼ cdom 8937 Fincfn 8939 |
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 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 |
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 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-om 7856 df-1o 8466 df-en 8940 df-dom 8941 df-fin 8943 |
This theorem is referenced by: ssdomfi 9199 |
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