Theorem f1domfi 9109

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 8912). (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 6799	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → ◡𝐹:ran 𝐹–1-1-onto→𝐴)
2		f1f 6731	. . . . . 6 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
3	2	frnd 6671	. . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → ran 𝐹 ⊆ 𝐵)
4		ssfi 9101	. . . . 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 6776	. . . . 5 ⊢ (◡𝐹:ran 𝐹–1-1-onto→𝐴 → ◡𝐹 Fn ran 𝐹)
7		fnfi 9106	. . . . 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 687	. . 3 ⊢ ((𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → ◡𝐹 ∈ Fin)
10		cnvfi 9104	. . . 4 ⊢ (◡𝐹 ∈ Fin → ◡◡𝐹 ∈ Fin)
11		f1rel 6734	. . . . . . 7 ⊢ (𝐹:𝐴–1-1→𝐵 → Rel 𝐹)
12		dfrel2 6148	. . . . . . 7 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹)
13	11, 12	sylib 218	. . . . . 6 ⊢ (𝐹:𝐴–1-1→𝐵 → ◡◡𝐹 = 𝐹)
14	13	eleq1d 2822	. . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → (◡◡𝐹 ∈ Fin ↔ 𝐹 ∈ Fin))
15	14	biimpac 478	. . . 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 8908	. . 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)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ⊆ wss 3902   class class class wbr 5099  ^◡ccnv 5624  ran crn 5626  Rel wrel 5630   Fn wfn 6488  –1-1→wf1 6490  –1-1-onto→wf1o 6492   ≼ cdom 8885  Fincfn 8887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-om 7811  df-1o 8399  df-en 8888  df-dom 8889  df-fin 8891

This theorem is referenced by:  ssdomfi  9124