Theorem fundmen 7084

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 5044	. . 3 ⊢ dom 𝐹 ∈ V
3	2	a1i 9	. 2 ⊢ (Fun 𝐹 → dom 𝐹 ∈ V)
4	1	a1i 9	. 2 ⊢ (Fun 𝐹 → 𝐹 ∈ V)
5		funfvop 5812	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹)
6	5	ex 115	. 2 ⊢ (Fun 𝐹 → (𝑥 ∈ dom 𝐹 → ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹))
7		funrel 5389	. . 3 ⊢ (Fun 𝐹 → Rel 𝐹)
8		elreldm 5003	. . . 4 ⊢ ((Rel 𝐹 ∧ 𝑦 ∈ 𝐹) → ∩ ∩ 𝑦 ∈ dom 𝐹)
9	8	ex 115	. . 3 ⊢ (Rel 𝐹 → (𝑦 ∈ 𝐹 → ∩ ∩ 𝑦 ∈ dom 𝐹))
10	7, 9	syl 14	. 2 ⊢ (Fun 𝐹 → (𝑦 ∈ 𝐹 → ∩ ∩ 𝑦 ∈ dom 𝐹))
11		df-rel 4776	. . . . . . . . 9 ⊢ (Rel 𝐹 ↔ 𝐹 ⊆ (V × V))
12	7, 11	sylib 122	. . . . . . . 8 ⊢ (Fun 𝐹 → 𝐹 ⊆ (V × V))
13	12	sselda 3248	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ 𝐹) → 𝑦 ∈ (V × V))
14		elvv 4832	. . . . . . 7 ⊢ (𝑦 ∈ (V × V) ↔ ∃𝑧∃𝑤 𝑦 = ⟨𝑧, 𝑤⟩)
15	13, 14	sylib 122	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ 𝐹) → ∃𝑧∃𝑤 𝑦 = ⟨𝑧, 𝑤⟩)
16		inteq 3968	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → ∩ 𝑦 = ∩ ⟨𝑧, 𝑤⟩)
17	16	inteqd 3970	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → ∩ ∩ 𝑦 = ∩ ∩ ⟨𝑧, 𝑤⟩)
18		vex 2824	. . . . . . . . . . . . . . . . 17 ⊢ 𝑧 ∈ V
19		vex 2824	. . . . . . . . . . . . . . . . 17 ⊢ 𝑤 ∈ V
20	18, 19	op1stb 4619	. . . . . . . . . . . . . . . 16 ⊢ ∩ ∩ ⟨𝑧, 𝑤⟩ = 𝑧
21	17, 20	eqtrdi 2287	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → ∩ ∩ 𝑦 = 𝑧)
22		eqeq1 2245	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ∩ ∩ 𝑦 → (𝑥 = 𝑧 ↔ ∩ ∩ 𝑦 = 𝑧))
23	21, 22	imbitrrid 156	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ∩ ∩ 𝑦 → (𝑦 = ⟨𝑧, 𝑤⟩ → 𝑥 = 𝑧))
24		opeq1 3899	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → ⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩)
25	23, 24	syl6 33	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∩ ∩ 𝑦 → (𝑦 = ⟨𝑧, 𝑤⟩ → ⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩))
26	25	imp 124	. . . . . . . . . . . 12 ⊢ ((𝑥 = ∩ ∩ 𝑦 ∧ 𝑦 = ⟨𝑧, 𝑤⟩) → ⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩)
27		eqeq2 2248	. . . . . . . . . . . . . 14 ⊢ (⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩ → (𝑦 = ⟨𝑥, 𝑤⟩ ↔ 𝑦 = ⟨𝑧, 𝑤⟩))
28	27	biimprcd 160	. . . . . . . . . . . . 13 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → (⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩ → 𝑦 = ⟨𝑥, 𝑤⟩))
29	28	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝑥 = ∩ ∩ 𝑦 ∧ 𝑦 = ⟨𝑧, 𝑤⟩) → (⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩ → 𝑦 = ⟨𝑥, 𝑤⟩))
30	26, 29	mpd 13	. . . . . . . . . . 11 ⊢ ((𝑥 = ∩ ∩ 𝑦 ∧ 𝑦 = ⟨𝑧, 𝑤⟩) → 𝑦 = ⟨𝑥, 𝑤⟩)
31	30	ancoms 268	. . . . . . . . . 10 ⊢ ((𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = ∩ ∩ 𝑦) → 𝑦 = ⟨𝑥, 𝑤⟩)
32	31	adantl 277	. . . . . . . . 9 ⊢ (((Fun 𝐹 ∧ 𝑦 ∈ 𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = ∩ ∩ 𝑦)) → 𝑦 = ⟨𝑥, 𝑤⟩)
33	30	eleq1d 2307	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = ∩ ∩ 𝑦 ∧ 𝑦 = ⟨𝑧, 𝑤⟩) → (𝑦 ∈ 𝐹 ↔ ⟨𝑥, 𝑤⟩ ∈ 𝐹))
34	33	adantl 277	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐹 ∧ (𝑥 = ∩ ∩ 𝑦 ∧ 𝑦 = ⟨𝑧, 𝑤⟩)) → (𝑦 ∈ 𝐹 ↔ ⟨𝑥, 𝑤⟩ ∈ 𝐹))
35		funopfv 5734	. . . . . . . . . . . . . . 15 ⊢ (Fun 𝐹 → (⟨𝑥, 𝑤⟩ ∈ 𝐹 → (𝐹‘𝑥) = 𝑤))
36	35	adantr 276	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐹 ∧ (𝑥 = ∩ ∩ 𝑦 ∧ 𝑦 = ⟨𝑧, 𝑤⟩)) → (⟨𝑥, 𝑤⟩ ∈ 𝐹 → (𝐹‘𝑥) = 𝑤))
37	34, 36	sylbid 150	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐹 ∧ (𝑥 = ∩ ∩ 𝑦 ∧ 𝑦 = ⟨𝑧, 𝑤⟩)) → (𝑦 ∈ 𝐹 → (𝐹‘𝑥) = 𝑤))
38	37	exp32 365	. . . . . . . . . . . 12 ⊢ (Fun 𝐹 → (𝑥 = ∩ ∩ 𝑦 → (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑦 ∈ 𝐹 → (𝐹‘𝑥) = 𝑤))))
39	38	com24 87	. . . . . . . . . . 11 ⊢ (Fun 𝐹 → (𝑦 ∈ 𝐹 → (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑥 = ∩ ∩ 𝑦 → (𝐹‘𝑥) = 𝑤))))
40	39	imp43 355	. . . . . . . . . 10 ⊢ (((Fun 𝐹 ∧ 𝑦 ∈ 𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = ∩ ∩ 𝑦)) → (𝐹‘𝑥) = 𝑤)
41	40	opeq2d 3906	. . . . . . . . 9 ⊢ (((Fun 𝐹 ∧ 𝑦 ∈ 𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = ∩ ∩ 𝑦)) → ⟨𝑥, (𝐹‘𝑥)⟩ = ⟨𝑥, 𝑤⟩)
42	32, 41	eqtr4d 2274	. . . . . . . 8 ⊢ (((Fun 𝐹 ∧ 𝑦 ∈ 𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = ∩ ∩ 𝑦)) → 𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩)
43	42	exp32 365	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑥 = ∩ ∩ 𝑦 → 𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩)))
44	43	exlimdvv 1953	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ 𝐹) → (∃𝑧∃𝑤 𝑦 = ⟨𝑧, 𝑤⟩ → (𝑥 = ∩ ∩ 𝑦 → 𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩)))
45	15, 44	mpd 13	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 = ∩ ∩ 𝑦 → 𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩))
46	45	adantrl 482	. . . 4 ⊢ ((Fun 𝐹 ∧ (𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝐹)) → (𝑥 = ∩ ∩ 𝑦 → 𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩))
47		inteq 3968	. . . . . . . . 9 ⊢ (𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩ → ∩ 𝑦 = ∩ ⟨𝑥, (𝐹‘𝑥)⟩)
48	47	inteqd 3970	. . . . . . . 8 ⊢ (𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩ → ∩ ∩ 𝑦 = ∩ ∩ ⟨𝑥, (𝐹‘𝑥)⟩)
49	48	adantl 277	. . . . . . 7 ⊢ (((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) ∧ 𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩) → ∩ ∩ 𝑦 = ∩ ∩ ⟨𝑥, (𝐹‘𝑥)⟩)
50		vex 2824	. . . . . . . . 9 ⊢ 𝑥 ∈ V
51		funfvex 5707	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ V)
52		op1stbg 4620	. . . . . . . . 9 ⊢ ((𝑥 ∈ V ∧ (𝐹‘𝑥) ∈ V) → ∩ ∩ ⟨𝑥, (𝐹‘𝑥)⟩ = 𝑥)
53	50, 51, 52	sylancr 418	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ∩ ∩ ⟨𝑥, (𝐹‘𝑥)⟩ = 𝑥)
54	53	adantr 276	. . . . . . 7 ⊢ (((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) ∧ 𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩) → ∩ ∩ ⟨𝑥, (𝐹‘𝑥)⟩ = 𝑥)
55	49, 54	eqtr2d 2272	. . . . . 6 ⊢ (((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) ∧ 𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩) → 𝑥 = ∩ ∩ 𝑦)
56	55	ex 115	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩ → 𝑥 = ∩ ∩ 𝑦))
57	56	adantrr 483	. . . 4 ⊢ ((Fun 𝐹 ∧ (𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝐹)) → (𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩ → 𝑥 = ∩ ∩ 𝑦))
58	46, 57	impbid 129	. . 3 ⊢ ((Fun 𝐹 ∧ (𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝐹)) → (𝑥 = ∩ ∩ 𝑦 ↔ 𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩))
59	58	ex 115	. 2 ⊢ (Fun 𝐹 → ((𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 = ∩ ∩ 𝑦 ↔ 𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩)))
60	3, 4, 6, 10, 59	en3d 7045	1 ⊢ (Fun 𝐹 → dom 𝐹 ≈ 𝐹)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402  ∃wex 1545   ∈ wcel 2209  Vcvv 2821   ⊆ wss 3220  ⟨cop 3708  ∩ cint 3965   class class class wbr 4125   × cxp 4767  dom cdm 4769  Rel wrel 4774  Fun wfun 5366  ‘cfv 5372   ≈ cen 7010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-en 7013

This theorem is referenced by:  fundmeng  7085