Theorem f1oeng 3298

Description: The domain and range of a one-to-one, onto function are equinumerous.

Assertion

Ref	Expression
f1oeng	⊢ (A ∈ C → (F:A–1-1-onto→B → A ≈ B))

Proof of Theorem f1oeng

Step	Hyp	Ref	Expression
1		fnex 2740	. . . 4 ⊢ (A ∈ C → (F Fn A → F ∈ V))
2		f1ofn 2801	. . . 4 ⊢ (F:A–1-1-onto→B → F Fn A)
3	1, 2	syl5 22	. . 3 ⊢ (A ∈ C → (F:A–1-1-onto→B → F ∈ V))
4		f1oeq1 2795	. . . 4 ⊢ (f = F → (f:A–1-1-onto→B ↔ F:A–1-1-onto→B))
5	4	cla4egv 1397	. . 3 ⊢ (F ∈ V → (F:A–1-1-onto→B → ∃f f:A–1-1-onto→B))
6	3, 5	syli 52	. 2 ⊢ (A ∈ C → (F:A–1-1-onto→B → ∃f f:A–1-1-onto→B))
7		brprc 2097	. . . . . 6 ⊢ (¬ B ∈ V → (A ≈ B ↔ A ≈ A))
8		enrefg 3294	. . . . . 6 ⊢ (A ∈ C → A ≈ A)
9	7, 8	syl5bir 184	. . . . 5 ⊢ (¬ B ∈ V → (A ∈ C → A ≈ B))
10	9	a1d 14	. . . 4 ⊢ (¬ B ∈ V → (∃f f:A–1-1-onto→B → (A ∈ C → A ≈ B)))
11	10	com3r 35	. . 3 ⊢ (A ∈ C → (¬ B ∈ V → (∃f f:A–1-1-onto→B → A ≈ B)))
12		breng 3280	. . . 4 ⊢ (B ∈ V → (A ≈ B ↔ ∃f f:A–1-1-onto→B))
13	12	biimprd 136	. . 3 ⊢ (B ∈ V → (∃f f:A–1-1-onto→B → A ≈ B))
14	11, 13	pm2.61d2 111	. 2 ⊢ (A ∈ C → (∃f f:A–1-1-onto→B → A ≈ B))
15	6, 14	syld 27	1 ⊢ (A ∈ C → (F:A–1-1-onto→B → A ≈ B))

Syntax hints:  ¬ wn 1   → wi 2  ∃wex 678   ∈ wcel 1092  Vcvv 1348   class class class wbr 2054   Fn wfn 2417  –1-1-onto→wf1o 2421   ≈ cen 3271

This theorem is referenced by:  f1oen 3301  en2d 3303

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-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-en 3274

metamath.org