Theorem f1imaenfi 9124

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 8956). (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 6790	. . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶))
2		f1oenfi 9108	. . . . 5 ⊢ ((𝐶 ∈ Fin ∧ (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶)) → 𝐶 ≈ (𝐹 “ 𝐶))
3		ensymfib 9113	. . . . . 6 ⊢ (𝐶 ∈ Fin → (𝐶 ≈ (𝐹 “ 𝐶) ↔ (𝐹 “ 𝐶) ≈ 𝐶))
4	3	adantr 480	. . . . 5 ⊢ ((𝐶 ∈ Fin ∧ (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶)) → (𝐶 ≈ (𝐹 “ 𝐶) ↔ (𝐹 “ 𝐶) ≈ 𝐶))
5	2, 4	mpbid 232	. . . 4 ⊢ ((𝐶 ∈ Fin ∧ (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶)) → (𝐹 “ 𝐶) ≈ 𝐶)
6	1, 5	sylan2 594	. . 3 ⊢ ((𝐶 ∈ Fin ∧ (𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴)) → (𝐹 “ 𝐶) ≈ 𝐶)
7	6	3impb 1115	. 2 ⊢ ((𝐶 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 “ 𝐶) ≈ 𝐶)
8	7	3coml 1128	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ Fin) → (𝐹 “ 𝐶) ≈ 𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   ∈ wcel 2114   ⊆ wss 3890   class class class wbr 5086   ↾ cres 5628   “ cima 5629  –1-1→wf1 6491  –1-1-onto→wf1o 6493   ≈ cen 8885  Fincfn 8888

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 5232  ax-nul 5242  ax-pr 5372  ax-un 7684

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 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-om 7813  df-1o 8400  df-en 8889  df-fin 8892

This theorem is referenced by:  phplem2  9134