Theorem breng 3280

Description: Equinumerosity relation.

Assertion

Ref	Expression
breng	⊢ (B ∈ C → (A ≈ B ↔ ∃f f:A–1-1-onto→B))

Distinct variable group(s):   A,f   B,f

Proof of Theorem breng

Step	Hyp	Ref	Expression
1		f1oeq2 2796	. . . . 5 ⊢ (x = A → (f:x–1-1-onto→y ↔ f:A–1-1-onto→y))
2	1	biexdv 936	. . . 4 ⊢ (x = A → (∃f f:x–1-1-onto→y ↔ ∃f f:A–1-1-onto→y))
3		f1oeq3 2797	. . . . 5 ⊢ (y = B → (f:A–1-1-onto→y ↔ f:A–1-1-onto→B))
4	3	biexdv 936	. . . 4 ⊢ (y = B → (∃f f:A–1-1-onto→y ↔ ∃f f:A–1-1-onto→B))
5		df-en 3274	. . . 4 ⊢ ≈ = {⟨x, y⟩∣∃f f:x–1-1-onto→y}
6	2, 4, 5	brabg 2116	. . 3 ⊢ ((A ∈ V ∧ B ∈ C) → (A ≈ B ↔ ∃f f:A–1-1-onto→B))
7	6	exp 291	. 2 ⊢ (A ∈ V → (B ∈ C → (A ≈ B ↔ ∃f f:A–1-1-onto→B)))
8		relen 3277	. . . . 5 ⊢ Rel ≈
9	8	brrelexi 2447	. . . 4 ⊢ (A ≈ B → A ∈ V)
10		f1ofn 2801	. . . . . 6 ⊢ (f:A–1-1-onto→B → f Fn A)
11		visset 1350	. . . . . . . 8 ⊢ f ∈ V
12		dmexg 2551	. . . . . . . 8 ⊢ (f ∈ V → dom f ∈ V)
13	11, 12	ax-mp 6	. . . . . . 7 ⊢ dom f ∈ V
14		fndm 2723	. . . . . . . 8 ⊢ (f Fn A → dom f = A)
15	14	eleq1d 1155	. . . . . . 7 ⊢ (f Fn A → (dom f ∈ V ↔ A ∈ V))
16	13, 15	mpbii 168	. . . . . 6 ⊢ (f Fn A → A ∈ V)
17	10, 16	syl 12	. . . . 5 ⊢ (f:A–1-1-onto→B → A ∈ V)
18	17	19.23aiv 952	. . . 4 ⊢ (∃f f:A–1-1-onto→B → A ∈ V)
19	9, 18	pm5.21ni 503	. . 3 ⊢ (¬ A ∈ V → (A ≈ B ↔ ∃f f:A–1-1-onto→B))
20	19	a1d 14	. 2 ⊢ (¬ A ∈ V → (B ∈ C → (A ≈ B ↔ ∃f f:A–1-1-onto→B)))
21	7, 20	pm2.61i 110	1 ⊢ (B ∈ C → (A ≈ B ↔ ∃f f:A–1-1-onto→B))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127  ∃wex 678   = wceq 1091   ∈ wcel 1092  Vcvv 1348   class class class wbr 2054  dom cdm 2410   Fn wfn 2417  –1-1-onto→wf1o 2421   ≈ cen 3271

This theorem is referenced by:  bren 3282  enrefg 3294  f1oeng 3298  unen 3338  ssfi 3430

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-clab 1093  df-cleq 1097  df-clel 1099  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-xp 2424  df-rel 2425  df-dm 2428  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-en 3274

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