Theorem breng 4366

Description: Equinumerosity relation.

Assertion

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

Distinct variable groups:   A,f   B,f

Proof of Theorem breng

Step	Hyp	Ref	Expression
1		f1oeq2 3680	. . . . 5 ⊢ (x = A → (f:x–1-1-onto→y ↔ f:A–1-1-onto→y))
2	1	exbidv 1278	. . . 4 ⊢ (x = A → (∃f f:x–1-1-onto→y ↔ ∃f f:A–1-1-onto→y))
3		f1oeq3 3681	. . . . 5 ⊢ (y = B → (f:A–1-1-onto→y ↔ f:A–1-1-onto→B))
4	3	exbidv 1278	. . . 4 ⊢ (y = B → (∃f f:A–1-1-onto→y ↔ ∃f f:A–1-1-onto→B))
5		df-en 4360	. . . 4 ⊢ ≈ = {⟨x, y⟩∣∃f f:x–1-1-onto→y}
6	2, 4, 5	brabg 2814	. . 3 ⊢ ((A ∈ V ⋀ B ∈ C) → (A ≈ B ↔ ∃f f:A–1-1-onto→B))
7	6	ex 373	. 2 ⊢ (A ∈ V → (B ∈ C → (A ≈ B ↔ ∃f f:A–1-1-onto→B)))
8		relen 4363	. . . . 5 ⊢ Rel ≈
9	8	brrelexi 3204	. . . 4 ⊢ (A ≈ B → A ∈ V)
10		f1ofn 3685	. . . . . 6 ⊢ (f:A–1-1-onto→B → f Fn A)
11		fndm 3583	. . . . . . 7 ⊢ (f Fn A → dom f = A)
12		visset 1810	. . . . . . . 8 ⊢ f ∈ V
13	12	dmex 3356	. . . . . . 7 ⊢ dom f ∈ V
14	11, 13	syl6eqelr 1555	. . . . . 6 ⊢ (f Fn A → A ∈ V)
15	10, 14	syl 10	. . . . 5 ⊢ (f:A–1-1-onto→B → A ∈ V)
16	15	19.23aiv 1294	. . . 4 ⊢ (∃f f:A–1-1-onto→B → A ∈ V)
17	9, 16	pm5.21ni 677	. . 3 ⊢ (¬ A ∈ V → (A ≈ B ↔ ∃f f:A–1-1-onto→B))
18	17	a1d 12	. 2 ⊢ (¬ A ∈ V → (B ∈ C → (A ≈ B ↔ ∃f f:A–1-1-onto→B)))
19	7, 18	pm2.61i 126	1 ⊢ (B ∈ C → (A ≈ B ↔ ∃f f:A–1-1-onto→B))

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146   = wceq 955   ∈ wcel 957  ∃wex 979  Vcvv 1808   class class class wbr 2615  dom cdm 3166   Fn wfn 3173  –1-1-onto→wf1o 3177   ≈ cen 4357

This theorem is referenced by:  bren 4368  enrefg 4380  f1oen2g 4384  unen 4423  ssfi 4524  homcard 10485

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-xp 3180  df-rel 3181  df-cnv 3182  df-dm 3184  df-rn 3185  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193  df-en 4360

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