Theorem f1domfi 9221

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 9023). (Contributed by BTernaryTau, 25-Sep-2024.)

Assertion

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

Proof of Theorem f1domfi

Step	Hyp	Ref	Expression
1		f1cnv 6868	. . . 4 ⊢ (𝐹:𝐴⟶1-1𝐵 → ◡𝐹:ran 𝐹⟶1-1onto𝐴)
2		f1f 6797	. . . . . 6 ⊢ (𝐹:𝐴⟶1-1𝐵 → 𝐹:𝐴⟶𝐵)
3	2	frnd 6737	. . . . 5 ⊢ (𝐹:𝐴⟶1-1𝐵 → ran 𝐹 ⊆ 𝐵)
4		ssfi 9213	. . . . 5 ⊢ ((𝐵 ∈ Fin ∧ ran 𝐹 ⊆ 𝐵) → ran 𝐹 ∈ Fin)
5	3, 4	sylan2 605	. . . 4 ⊢ ((𝐵 ∈ Fin ∧ 𝐹:𝐴⟶1-1𝐵) → ran 𝐹 ∈ Fin)
6		f1ofn 6844	. . . . 5 ⊢ (◡𝐹:ran 𝐹⟶1-1onto𝐴 → ◡𝐹 Fn ran 𝐹)
7		fnfi 9218	. . . . 5 ⊢ ((◡𝐹 Fn ran 𝐹 ∧ ran 𝐹 ∈ Fin) → ◡𝐹 ∈ Fin)
8	6, 7	sylan 592	. . . 4 ⊢ ((◡𝐹:ran 𝐹⟶1-1onto𝐴 ∧ ran 𝐹 ∈ Fin) → ◡𝐹 ∈ Fin)
9	1, 5, 8	syl2an2 699	. . 3 ⊢ ((𝐵 ∈ Fin ∧ 𝐹:𝐴⟶1-1𝐵) → ◡𝐹 ∈ Fin)
10		cnvfi 9216	. . . 4 ⊢ (◡𝐹 ∈ Fin → ◡◡𝐹 ∈ Fin)
11		f1rel 6801	. . . . . . 7 ⊢ (𝐹:𝐴⟶1-1𝐵 → Rel 𝐹)
12		dfrel2 6192	. . . . . . 7 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹)
13	11, 12	sylib 221	. . . . . 6 ⊢ (𝐹:𝐴⟶1-1𝐵 → ◡◡𝐹 = 𝐹)
14	13	eleq1d 2849	. . . . 5 ⊢ (𝐹:𝐴⟶1-1𝐵 → (◡◡𝐹 ∈ Fin ↔ 𝐹 ∈ Fin))
15	14	biimpac 484	. . . 4 ⊢ ((◡◡𝐹 ∈ Fin ∧ 𝐹:𝐴⟶1-1𝐵) → 𝐹 ∈ Fin)
16	10, 15	sylan 592	. . 3 ⊢ ((◡𝐹 ∈ Fin ∧ 𝐹:𝐴⟶1-1𝐵) → 𝐹 ∈ Fin)
17	9, 16	sylancom 600	. 2 ⊢ ((𝐵 ∈ Fin ∧ 𝐹:𝐴⟶1-1𝐵) → 𝐹 ∈ Fin)
18		f1dom3g 9019	. . 3 ⊢ ((𝐹 ∈ Fin ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴⟶1-1𝐵) → 𝐴 ≼ 𝐵)
19	18	3expib 1140	. 2 ⊢ (𝐹 ∈ Fin → ((𝐵 ∈ Fin ∧ 𝐹:𝐴⟶1-1𝐵) → 𝐴 ≼ 𝐵))
20	17, 19	mpcom 39	1 ⊢ ((𝐵 ∈ Fin ∧ 𝐹:𝐴⟶1-1𝐵) → 𝐴 ≼ 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900   class class class wbr 5106  ^◡ccnv 5654  ran crn 5656  Rel wrel 5660   Fn wfn 6550  ⟶1-1wf1 6552  ⟶1-1ontowf1o 6554   ≼ cdom 8996  Fincfn 8998

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-om 7899  df-1o 8501  df-en 8999  df-dom 9000  df-fin 9002

This theorem is used by:  ssdomfi  9236