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Theorem f1domfi 9219
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 9011). (Contributed by BTernaryTau, 25-Sep-2024.)
Assertion
Ref Expression
f1domfi ((𝐵 ∈ Fin ∧ 𝐹:𝐴1-1𝐵) → 𝐴𝐵)

Proof of Theorem f1domfi
StepHypRef Expression
1 f1cnv 6873 . . . 4 (𝐹:𝐴1-1𝐵𝐹:ran 𝐹1-1-onto𝐴)
2 f1f 6805 . . . . . 6 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
32frnd 6745 . . . . 5 (𝐹:𝐴1-1𝐵 → ran 𝐹𝐵)
4 ssfi 9212 . . . . 5 ((𝐵 ∈ Fin ∧ ran 𝐹𝐵) → ran 𝐹 ∈ Fin)
53, 4sylan2 593 . . . 4 ((𝐵 ∈ Fin ∧ 𝐹:𝐴1-1𝐵) → ran 𝐹 ∈ Fin)
6 f1ofn 6850 . . . . 5 (𝐹:ran 𝐹1-1-onto𝐴𝐹 Fn ran 𝐹)
7 fnfi 9216 . . . . 5 ((𝐹 Fn ran 𝐹 ∧ ran 𝐹 ∈ Fin) → 𝐹 ∈ Fin)
86, 7sylan 580 . . . 4 ((𝐹:ran 𝐹1-1-onto𝐴 ∧ ran 𝐹 ∈ Fin) → 𝐹 ∈ Fin)
91, 5, 8syl2an2 686 . . 3 ((𝐵 ∈ Fin ∧ 𝐹:𝐴1-1𝐵) → 𝐹 ∈ Fin)
10 cnvfi 9215 . . . 4 (𝐹 ∈ Fin → 𝐹 ∈ Fin)
11 f1rel 6808 . . . . . . 7 (𝐹:𝐴1-1𝐵 → Rel 𝐹)
12 dfrel2 6211 . . . . . . 7 (Rel 𝐹𝐹 = 𝐹)
1311, 12sylib 218 . . . . . 6 (𝐹:𝐴1-1𝐵𝐹 = 𝐹)
1413eleq1d 2824 . . . . 5 (𝐹:𝐴1-1𝐵 → (𝐹 ∈ Fin ↔ 𝐹 ∈ Fin))
1514biimpac 478 . . . 4 ((𝐹 ∈ Fin ∧ 𝐹:𝐴1-1𝐵) → 𝐹 ∈ Fin)
1610, 15sylan 580 . . 3 ((𝐹 ∈ Fin ∧ 𝐹:𝐴1-1𝐵) → 𝐹 ∈ Fin)
179, 16sylancom 588 . 2 ((𝐵 ∈ Fin ∧ 𝐹:𝐴1-1𝐵) → 𝐹 ∈ Fin)
18 f1dom3g 9007 . . 3 ((𝐹 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴1-1𝐵) → 𝐴𝐵)
19183expib 1121 . 2 (𝐹 ∈ Fin → ((𝐵 ∈ Fin ∧ 𝐹:𝐴1-1𝐵) → 𝐴𝐵))
2017, 19mpcom 38 1 ((𝐵 ∈ Fin ∧ 𝐹:𝐴1-1𝐵) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wss 3963   class class class wbr 5148  ccnv 5688  ran crn 5690  Rel wrel 5694   Fn wfn 6558  1-1wf1 6560  1-1-ontowf1o 6562  cdom 8982  Fincfn 8984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-1o 8505  df-en 8985  df-dom 8986  df-fin 8988
This theorem is referenced by:  ssdomfi  9234
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