Theorem f1domfi 9247

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 9032). (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 6886	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → ◡𝐹:ran 𝐹–1-1-onto→𝐴)
2		f1f 6817	. . . . . 6 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
3	2	frnd 6755	. . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → ran 𝐹 ⊆ 𝐵)
4		ssfi 9240	. . . . 5 ⊢ ((𝐵 ∈ Fin ∧ ran 𝐹 ⊆ 𝐵) → ran 𝐹 ∈ Fin)
5	3, 4	sylan2 592	. . . 4 ⊢ ((𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → ran 𝐹 ∈ Fin)
6		f1ofn 6863	. . . . 5 ⊢ (◡𝐹:ran 𝐹–1-1-onto→𝐴 → ◡𝐹 Fn ran 𝐹)
7		fnfi 9244	. . . . 5 ⊢ ((◡𝐹 Fn ran 𝐹 ∧ ran 𝐹 ∈ Fin) → ◡𝐹 ∈ Fin)
8	6, 7	sylan 579	. . . 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 9243	. . . 4 ⊢ (◡𝐹 ∈ Fin → ◡◡𝐹 ∈ Fin)
11		f1rel 6820	. . . . . . 7 ⊢ (𝐹:𝐴–1-1→𝐵 → Rel 𝐹)
12		dfrel2 6220	. . . . . . 7 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹)
13	11, 12	sylib 218	. . . . . 6 ⊢ (𝐹:𝐴–1-1→𝐵 → ◡◡𝐹 = 𝐹)
14	13	eleq1d 2829	. . . . 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 579	. . 3 ⊢ ((◡𝐹 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹 ∈ Fin)
17	9, 16	sylancom 587	. 2 ⊢ ((𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹 ∈ Fin)
18		f1dom3g 9027	. . 3 ⊢ ((𝐹 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵)
19	18	3expib 1122	. 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 1537   ∈ wcel 2108   ⊆ wss 3976   class class class wbr 5166  ^◡ccnv 5699  ran crn 5701  Rel wrel 5705   Fn wfn 6568  –1-1→wf1 6570  –1-1-onto→wf1o 6572   ≼ cdom 9001  Fincfn 9003

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-om 7904  df-1o 8522  df-en 9004  df-dom 9005  df-fin 9007

This theorem is referenced by:  ssdomfi  9262