Theorem f1domfi 9196

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 8998). (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 6849	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → ◡𝐹:ran 𝐹–1-1-onto→𝐴)
2		f1f 6778	. . . . . 6 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
3	2	frnd 6718	. . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → ran 𝐹 ⊆ 𝐵)
4		ssfi 9188	. . . . 5 ⊢ ((𝐵 ∈ Fin ∧ ran 𝐹 ⊆ 𝐵) → ran 𝐹 ∈ Fin)
5	3, 4	sylan2 605	. . . 4 ⊢ ((𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → ran 𝐹 ∈ Fin)
6		f1ofn 6825	. . . . 5 ⊢ (◡𝐹:ran 𝐹–1-1-onto→𝐴 → ◡𝐹 Fn ran 𝐹)
7		fnfi 9193	. . . . 5 ⊢ ((◡𝐹 Fn ran 𝐹 ∧ ran 𝐹 ∈ Fin) → ◡𝐹 ∈ Fin)
8	6, 7	sylan 592	. . . 4 ⊢ ((◡𝐹:ran 𝐹–1-1-onto→𝐴 ∧ ran 𝐹 ∈ Fin) → ◡𝐹 ∈ Fin)
9	1, 5, 8	syl2an2 699	. . 3 ⊢ ((𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → ◡𝐹 ∈ Fin)
10		cnvfi 9191	. . . 4 ⊢ (◡𝐹 ∈ Fin → ◡◡𝐹 ∈ Fin)
11		f1rel 6782	. . . . . . 7 ⊢ (𝐹:𝐴–1-1→𝐵 → Rel 𝐹)
12		dfrel2 6180	. . . . . . 7 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹)
13	11, 12	sylib 221	. . . . . 6 ⊢ (𝐹:𝐴–1-1→𝐵 → ◡◡𝐹 = 𝐹)
14	13	eleq1d 2845	. . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → (◡◡𝐹 ∈ Fin ↔ 𝐹 ∈ Fin))
15	14	biimpac 484	. . . 4 ⊢ ((◡◡𝐹 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹 ∈ Fin)
16	10, 15	sylan 592	. . 3 ⊢ ((◡𝐹 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹 ∈ Fin)
17	9, 16	sylancom 600	. 2 ⊢ ((𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹 ∈ Fin)
18		f1dom3g 8994	. . 3 ⊢ ((𝐹 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵)
19	18	3expib 1140	. 2 ⊢ (𝐹 ∈ Fin → ((𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵))
20	17, 19	mpcom 39	1 ⊢ ((𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ⊆ wss 3895   class class class wbr 5102  ^◡ccnv 5649  ran crn 5651  Rel wrel 5655   Fn wfn 6533  –1-1→wf1 6535  –1-1-onto→wf1o 6537   ≼ cdom 8971  Fincfn 8973

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-om 7878  df-1o 8476  df-en 8974  df-dom 8975  df-fin 8977

This theorem is used by:  ssdomfi  9211