Theorem f1domfi 8967

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 8760). (Contributed by BTernaryTau, 25-Sep-2024.)

Assertion

Ref	Expression
f1domfi	⊢ ((𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵)

Proof of Theorem f1domfi

Step	Hyp	Ref	Expression
1		f1cnv 6740	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → ◡𝐹:ran 𝐹–1-1-onto→𝐴)
2		f1f 6670	. . . . . 6 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
3	2	frnd 6608	. . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → ran 𝐹 ⊆ 𝐵)
4		ssfi 8956	. . . . 5 ⊢ ((𝐵 ∈ Fin ∧ ran 𝐹 ⊆ 𝐵) → ran 𝐹 ∈ Fin)
5	3, 4	sylan2 593	. . . 4 ⊢ ((𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → ran 𝐹 ∈ Fin)
6		f1ofn 6717	. . . . 5 ⊢ (◡𝐹:ran 𝐹–1-1-onto→𝐴 → ◡𝐹 Fn ran 𝐹)
7		fnfi 8964	. . . . 5 ⊢ ((◡𝐹 Fn ran 𝐹 ∧ ran 𝐹 ∈ Fin) → ◡𝐹 ∈ Fin)
8	6, 7	sylan 580	. . . 4 ⊢ ((◡𝐹:ran 𝐹–1-1-onto→𝐴 ∧ ran 𝐹 ∈ Fin) → ◡𝐹 ∈ Fin)
9	1, 5, 8	syl2an2 683	. . 3 ⊢ ((𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → ◡𝐹 ∈ Fin)
10		cnvfi 8963	. . . 4 ⊢ (◡𝐹 ∈ Fin → ◡◡𝐹 ∈ Fin)
11		f1rel 6673	. . . . . . 7 ⊢ (𝐹:𝐴–1-1→𝐵 → Rel 𝐹)
12		dfrel2 6092	. . . . . . 7 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹)
13	11, 12	sylib 217	. . . . . 6 ⊢ (𝐹:𝐴–1-1→𝐵 → ◡◡𝐹 = 𝐹)
14	13	eleq1d 2823	. . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → (◡◡𝐹 ∈ Fin ↔ 𝐹 ∈ Fin))
15	14	biimpac 479	. . . 4 ⊢ ((◡◡𝐹 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹 ∈ Fin)
16	10, 15	sylan 580	. . 3 ⊢ ((◡𝐹 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹 ∈ Fin)
17	9, 16	sylancom 588	. 2 ⊢ ((𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹 ∈ Fin)
18		f1dom3g 8755	. . 3 ⊢ ((𝐹 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵)
19	18	3expib 1121	. 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)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ⊆ wss 3887   class class class wbr 5074  ^◡ccnv 5588  ran crn 5590  Rel wrel 5594   Fn wfn 6428  –1-1→wf1 6430  –1-1-onto→wf1o 6432   ≼ cdom 8731  Fincfn 8733

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-om 7713  df-1o 8297  df-en 8734  df-dom 8735  df-fin 8737

This theorem is referenced by:  ssdomfi  8982