Theorem fundmen 9035

Description: A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98. (Contributed by NM, 28-Jul-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)

Hypothesis

Ref	Expression
fundmen.1	⊢ 𝐹 ∈ V

Assertion

Ref	Expression
fundmen	⊢ (Fun 𝐹 → dom 𝐹 ≈ 𝐹)

Proof of Theorem fundmen

Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fundmen.1	. . . 4 ⊢ 𝐹 ∈ V
2	1	dmex 7912	. . 3 ⊢ dom 𝐹 ∈ V
3	2	a1i 11	. 2 ⊢ (Fun 𝐹 → dom 𝐹 ∈ V)
4	1	a1i 11	. 2 ⊢ (Fun 𝐹 → 𝐹 ∈ V)
5		funfvop 7049	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹)
6	5	ex 418	. 2 ⊢ (Fun 𝐹 → (𝑥 ∈ dom 𝐹 → ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹))
7		funrel 6557	. . 3 ⊢ (Fun 𝐹 → Rel 𝐹)
8		elreldm 5927	. . . 4 ⊢ ((Rel 𝐹 ∧ 𝑦 ∈ 𝐹) → ∩ ∩ 𝑦 ∈ dom 𝐹)
9	8	ex 418	. . 3 ⊢ (Rel 𝐹 → (𝑦 ∈ 𝐹 → ∩ ∩ 𝑦 ∈ dom 𝐹))
10	7, 9	syl 18	. 2 ⊢ (Fun 𝐹 → (𝑦 ∈ 𝐹 → ∩ ∩ 𝑦 ∈ dom 𝐹))
11		df-rel 5670	. . . . . . . . 9 ⊢ (Rel 𝐹 ↔ 𝐹 ⊆ (V × V))
12	7, 11	sylib 221	. . . . . . . 8 ⊢ (Fun 𝐹 → 𝐹 ⊆ (V × V))
13	12	sselda 3938	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ 𝐹) → 𝑦 ∈ (V × V))
14		elvv 5738	. . . . . . 7 ⊢ (𝑦 ∈ (V × V) ↔ ∃𝑧∃𝑤 𝑦 = ⟨𝑧, 𝑤⟩)
15	13, 14	sylib 221	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ 𝐹) → ∃𝑧∃𝑤 𝑦 = ⟨𝑧, 𝑤⟩)
16		inteq 4917	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → ∩ 𝑦 = ∩ ⟨𝑧, 𝑤⟩)
17	16	inteqd 4919	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → ∩ ∩ 𝑦 = ∩ ∩ ⟨𝑧, 𝑤⟩)
18		vex 3461	. . . . . . . . . . . . . . . . 17 ⊢ 𝑧 ∈ V
19		vex 3461	. . . . . . . . . . . . . . . . 17 ⊢ 𝑤 ∈ V
20	18, 19	op1stb 5455	. . . . . . . . . . . . . . . 16 ⊢ ∩ ∩ ⟨𝑧, 𝑤⟩ = 𝑧
21	17, 20	eqtrdi 2816	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → ∩ ∩ 𝑦 = 𝑧)
22		eqeq1 2769	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ∩ ∩ 𝑦 → (𝑥 = 𝑧 ↔ ∩ ∩ 𝑦 = 𝑧))
23	21, 22	imbitrrid 249	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ∩ ∩ 𝑦 → (𝑦 = ⟨𝑧, 𝑤⟩ → 𝑥 = 𝑧))
24		opeq1 4840	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → ⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩)
25	23, 24	syl6 36	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∩ ∩ 𝑦 → (𝑦 = ⟨𝑧, 𝑤⟩ → ⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩))
26	25	imp 412	. . . . . . . . . . . 12 ⊢ ((𝑥 = ∩ ∩ 𝑦 ∧ 𝑦 = ⟨𝑧, 𝑤⟩) → ⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩)
27		eqeq2 2777	. . . . . . . . . . . . . 14 ⊢ (⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩ → (𝑦 = ⟨𝑥, 𝑤⟩ ↔ 𝑦 = ⟨𝑧, 𝑤⟩))
28	27	biimprcd 253	. . . . . . . . . . . . 13 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → (⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩ → 𝑦 = ⟨𝑥, 𝑤⟩))
29	28	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝑥 = ∩ ∩ 𝑦 ∧ 𝑦 = ⟨𝑧, 𝑤⟩) → (⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩ → 𝑦 = ⟨𝑥, 𝑤⟩))
30	26, 29	mpd 16	. . . . . . . . . . 11 ⊢ ((𝑥 = ∩ ∩ 𝑦 ∧ 𝑦 = ⟨𝑧, 𝑤⟩) → 𝑦 = ⟨𝑥, 𝑤⟩)
31	30	ancoms 464	. . . . . . . . . 10 ⊢ ((𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = ∩ ∩ 𝑦) → 𝑦 = ⟨𝑥, 𝑤⟩)
32	31	adantl 487	. . . . . . . . 9 ⊢ (((Fun 𝐹 ∧ 𝑦 ∈ 𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = ∩ ∩ 𝑦)) → 𝑦 = ⟨𝑥, 𝑤⟩)
33	30	eleq1d 2850	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = ∩ ∩ 𝑦 ∧ 𝑦 = ⟨𝑧, 𝑤⟩) → (𝑦 ∈ 𝐹 ↔ ⟨𝑥, 𝑤⟩ ∈ 𝐹))
34	33	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐹 ∧ (𝑥 = ∩ ∩ 𝑦 ∧ 𝑦 = ⟨𝑧, 𝑤⟩)) → (𝑦 ∈ 𝐹 ↔ ⟨𝑥, 𝑤⟩ ∈ 𝐹))
35		funopfv 6934	. . . . . . . . . . . . . . 15 ⊢ (Fun 𝐹 → (⟨𝑥, 𝑤⟩ ∈ 𝐹 → (𝐹‘𝑥) = 𝑤))
36	35	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐹 ∧ (𝑥 = ∩ ∩ 𝑦 ∧ 𝑦 = ⟨𝑧, 𝑤⟩)) → (⟨𝑥, 𝑤⟩ ∈ 𝐹 → (𝐹‘𝑥) = 𝑤))
37	34, 36	sylbid 243	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐹 ∧ (𝑥 = ∩ ∩ 𝑦 ∧ 𝑦 = ⟨𝑧, 𝑤⟩)) → (𝑦 ∈ 𝐹 → (𝐹‘𝑥) = 𝑤))
38	37	exp32 426	. . . . . . . . . . . 12 ⊢ (Fun 𝐹 → (𝑥 = ∩ ∩ 𝑦 → (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑦 ∈ 𝐹 → (𝐹‘𝑥) = 𝑤))))
39	38	com24 96	. . . . . . . . . . 11 ⊢ (Fun 𝐹 → (𝑦 ∈ 𝐹 → (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑥 = ∩ ∩ 𝑦 → (𝐹‘𝑥) = 𝑤))))
40	39	imp43 433	. . . . . . . . . 10 ⊢ (((Fun 𝐹 ∧ 𝑦 ∈ 𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = ∩ ∩ 𝑦)) → (𝐹‘𝑥) = 𝑤)
41	40	opeq2d 4847	. . . . . . . . 9 ⊢ (((Fun 𝐹 ∧ 𝑦 ∈ 𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = ∩ ∩ 𝑦)) → ⟨𝑥, (𝐹‘𝑥)⟩ = ⟨𝑥, 𝑤⟩)
42	32, 41	eqtr4d 2803	. . . . . . . 8 ⊢ (((Fun 𝐹 ∧ 𝑦 ∈ 𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = ∩ ∩ 𝑦)) → 𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩)
43	42	exp32 426	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑥 = ∩ ∩ 𝑦 → 𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩)))
44	43	exlimdvv 1967	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ 𝐹) → (∃𝑧∃𝑤 𝑦 = ⟨𝑧, 𝑤⟩ → (𝑥 = ∩ ∩ 𝑦 → 𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩)))
45	15, 44	mpd 16	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 = ∩ ∩ 𝑦 → 𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩))
46	45	adantrl 729	. . . 4 ⊢ ((Fun 𝐹 ∧ (𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝐹)) → (𝑥 = ∩ ∩ 𝑦 → 𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩))
47		inteq 4917	. . . . . 6 ⊢ (𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩ → ∩ 𝑦 = ∩ ⟨𝑥, (𝐹‘𝑥)⟩)
48	47	inteqd 4919	. . . . 5 ⊢ (𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩ → ∩ ∩ 𝑦 = ∩ ∩ ⟨𝑥, (𝐹‘𝑥)⟩)
49		vex 3461	. . . . . 6 ⊢ 𝑥 ∈ V
50		fvex 6898	. . . . . 6 ⊢ (𝐹‘𝑥) ∈ V
51	49, 50	op1stb 5455	. . . . 5 ⊢ ∩ ∩ ⟨𝑥, (𝐹‘𝑥)⟩ = 𝑥
52	48, 51	eqtr2di 2817	. . . 4 ⊢ (𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩ → 𝑥 = ∩ ∩ 𝑦)
53	46, 52	impbid1 228	. . 3 ⊢ ((Fun 𝐹 ∧ (𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝐹)) → (𝑥 = ∩ ∩ 𝑦 ↔ 𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩))
54	53	ex 418	. 2 ⊢ (Fun 𝐹 → ((𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 = ∩ ∩ 𝑦 ↔ 𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩)))
55	3, 4, 6, 10, 54	en3d 8992	1 ⊢ (Fun 𝐹 → dom 𝐹 ≈ 𝐹)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146  Vcvv 3457   ⊆ wss 3906  ⟨cop 4597  ∩ cint 4914   class class class wbr 5111   × cxp 5661  dom cdm 5663  Rel wrel 5668  Fun wfun 6534  ‘cfv 6540   ≈ cen 8946

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-en 8950

This theorem is used by:  fundmeng  9036  infmap2  10216  heicant  38365