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Theorem f1domfi 9131
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 8915). (Contributed by BTernaryTau, 25-Sep-2024.)
Assertion
Ref Expression
f1domfi ((𝐵 ∈ Fin ∧ 𝐹:𝐴1-1𝐵) → 𝐴𝐵)

Proof of Theorem f1domfi
StepHypRef Expression
1 f1cnv 6809 . . . 4 (𝐹:𝐴1-1𝐵𝐹:ran 𝐹1-1-onto𝐴)
2 f1f 6739 . . . . . 6 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
32frnd 6677 . . . . 5 (𝐹:𝐴1-1𝐵 → ran 𝐹𝐵)
4 ssfi 9120 . . . . 5 ((𝐵 ∈ Fin ∧ ran 𝐹𝐵) → ran 𝐹 ∈ Fin)
53, 4sylan2 594 . . . 4 ((𝐵 ∈ Fin ∧ 𝐹:𝐴1-1𝐵) → ran 𝐹 ∈ Fin)
6 f1ofn 6786 . . . . 5 (𝐹:ran 𝐹1-1-onto𝐴𝐹 Fn ran 𝐹)
7 fnfi 9128 . . . . 5 ((𝐹 Fn ran 𝐹 ∧ ran 𝐹 ∈ Fin) → 𝐹 ∈ Fin)
86, 7sylan 581 . . . 4 ((𝐹:ran 𝐹1-1-onto𝐴 ∧ ran 𝐹 ∈ Fin) → 𝐹 ∈ Fin)
91, 5, 8syl2an2 685 . . 3 ((𝐵 ∈ Fin ∧ 𝐹:𝐴1-1𝐵) → 𝐹 ∈ Fin)
10 cnvfi 9127 . . . 4 (𝐹 ∈ Fin → 𝐹 ∈ Fin)
11 f1rel 6742 . . . . . . 7 (𝐹:𝐴1-1𝐵 → Rel 𝐹)
12 dfrel2 6142 . . . . . . 7 (Rel 𝐹𝐹 = 𝐹)
1311, 12sylib 217 . . . . . 6 (𝐹:𝐴1-1𝐵𝐹 = 𝐹)
1413eleq1d 2819 . . . . 5 (𝐹:𝐴1-1𝐵 → (𝐹 ∈ Fin ↔ 𝐹 ∈ Fin))
1514biimpac 480 . . . 4 ((𝐹 ∈ Fin ∧ 𝐹:𝐴1-1𝐵) → 𝐹 ∈ Fin)
1610, 15sylan 581 . . 3 ((𝐹 ∈ Fin ∧ 𝐹:𝐴1-1𝐵) → 𝐹 ∈ Fin)
179, 16sylancom 589 . 2 ((𝐵 ∈ Fin ∧ 𝐹:𝐴1-1𝐵) → 𝐹 ∈ Fin)
18 f1dom3g 8910 . . 3 ((𝐹 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴1-1𝐵) → 𝐴𝐵)
19183expib 1123 . 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 397   = wceq 1542  wcel 2107  wss 3911   class class class wbr 5106  ccnv 5633  ran crn 5635  Rel wrel 5639   Fn wfn 6492  1-1wf1 6494  1-1-ontowf1o 6496  cdom 8884  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-dom 8888  df-fin 8890
This theorem is referenced by:  ssdomfi  9146
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