Theorem f1imaen 3327

Description: A one-to-one function's image under a subset of its domain is equinumerous to the subset.

Hypothesis

Ref	Expression
f1imaen.1	⊢ C ∈ V

Assertion

Ref	Expression
f1imaen	⊢ ((F:A–1-1→B ∧ C ⊆ A) → (F “ C) ≈ C)

Proof of Theorem f1imaen

Step	Hyp	Ref	Expression
1		f1ores 2813	. 2 ⊢ ((F:A–1-1→B ∧ C ⊆ A) → (F ↾ C):C–1-1-onto→(F “ C))
2		f1imaen.1	. . . 4 ⊢ C ∈ V
3	2	f1oen 3301	. . 3 ⊢ ((F ↾ C):C–1-1-onto→(F “ C) → C ≈ (F “ C))
4		f1ofo 2806	. . . 4 ⊢ ((F ↾ C):C–1-1-onto→(F “ C) → (F ↾ C):C–onto→(F “ C))
5		fornex 2793	. . . . 5 ⊢ (C ∈ V → ((F ↾ C):C–onto→(F “ C) → (F “ C) ∈ V))
6	2, 5	ax-mp 6	. . . 4 ⊢ ((F ↾ C):C–onto→(F “ C) → (F “ C) ∈ V)
7		ensymg 3316	. . . 4 ⊢ ((F “ C) ∈ V → (C ≈ (F “ C) → (F “ C) ≈ C))
8	4, 6, 7	3syl 21	. . 3 ⊢ ((F ↾ C):C–1-1-onto→(F “ C) → (C ≈ (F “ C) → (F “ C) ≈ C))
9	3, 8	mpd 46	. 2 ⊢ ((F ↾ C):C–1-1-onto→(F “ C) → (F “ C) ≈ C)
10	1, 9	syl 12	1 ⊢ ((F:A–1-1→B ∧ C ⊆ A) → (F “ C) ≈ C)

Syntax hints:   → wi 2   ∧ wa 196   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487   class class class wbr 2054   ↾ cres 2412   “ cima 2413  –1-1→wf1 2419  –onto→wfo 2420  –1-1-onto→wf1o 2421   ≈ cen 3271

This theorem is referenced by:  ssenen 3399

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-er 3200  df-en 3274

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