Theorem f1imaenfi 9188

Description: If a function is one-to-one, then the image of a finite subset of its domain under it is equinumerous to the subset. This theorem is proved without using the Axiom of Replacement or the Axiom of Power Sets (unlike f1imaeng 9019). (Contributed by BTernaryTau, 29-Sep-2024.)

Assertion

Ref	Expression
f1imaenfi	⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ Fin) → (𝐹 “ 𝐶) ≈ 𝐶)

Proof of Theorem f1imaenfi

Step	Hyp	Ref	Expression
1		f1ores 6827	. . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶))
2		f1oenfi 9172	. . . . 5 ⊢ ((𝐶 ∈ Fin ∧ (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶)) → 𝐶 ≈ (𝐹 “ 𝐶))
3		ensymfib 9177	. . . . . 6 ⊢ (𝐶 ∈ Fin → (𝐶 ≈ (𝐹 “ 𝐶) ↔ (𝐹 “ 𝐶) ≈ 𝐶))
4	3	adantr 486	. . . . 5 ⊢ ((𝐶 ∈ Fin ∧ (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶)) → (𝐶 ≈ (𝐹 “ 𝐶) ↔ (𝐹 “ 𝐶) ≈ 𝐶))
5	2, 4	mpbid 235	. . . 4 ⊢ ((𝐶 ∈ Fin ∧ (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶)) → (𝐹 “ 𝐶) ≈ 𝐶)
6	1, 5	sylan2 605	. . 3 ⊢ ((𝐶 ∈ Fin ∧ (𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴)) → (𝐹 “ 𝐶) ≈ 𝐶)
7	6	3impb 1132	. 2 ⊢ ((𝐶 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 “ 𝐶) ≈ 𝐶)
8	7	3coml 1145	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ Fin) → (𝐹 “ 𝐶) ≈ 𝐶)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   ∈ wcel 2145   ⊆ wss 3898   class class class wbr 5102   ↾ cres 5649   “ cima 5650  –1-1→wf1 6524  –1-1-onto→wf1o 6526   ≈ cen 8948  Fincfn 8951

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7734

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 2100  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-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  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 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-om 7861  df-1o 8454  df-en 8952  df-fin 8955

This theorem is used by:  phplem2  9198