Theorem fundmen 8775

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 7732	. . 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 6909	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹)
6	5	ex 412	. 2 ⊢ (Fun 𝐹 → (𝑥 ∈ dom 𝐹 → ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹))
7		funrel 6435	. . 3 ⊢ (Fun 𝐹 → Rel 𝐹)
8		elreldm 5833	. . . 4 ⊢ ((Rel 𝐹 ∧ 𝑦 ∈ 𝐹) → ∩ ∩ 𝑦 ∈ dom 𝐹)
9	8	ex 412	. . 3 ⊢ (Rel 𝐹 → (𝑦 ∈ 𝐹 → ∩ ∩ 𝑦 ∈ dom 𝐹))
10	7, 9	syl 17	. 2 ⊢ (Fun 𝐹 → (𝑦 ∈ 𝐹 → ∩ ∩ 𝑦 ∈ dom 𝐹))
11		df-rel 5587	. . . . . . . . 9 ⊢ (Rel 𝐹 ↔ 𝐹 ⊆ (V × V))
12	7, 11	sylib 217	. . . . . . . 8 ⊢ (Fun 𝐹 → 𝐹 ⊆ (V × V))
13	12	sselda 3917	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ 𝐹) → 𝑦 ∈ (V × V))
14		elvv 5652	. . . . . . 7 ⊢ (𝑦 ∈ (V × V) ↔ ∃𝑧∃𝑤 𝑦 = ⟨𝑧, 𝑤⟩)
15	13, 14	sylib 217	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ 𝐹) → ∃𝑧∃𝑤 𝑦 = ⟨𝑧, 𝑤⟩)
16		inteq 4879	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → ∩ 𝑦 = ∩ ⟨𝑧, 𝑤⟩)
17	16	inteqd 4881	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → ∩ ∩ 𝑦 = ∩ ∩ ⟨𝑧, 𝑤⟩)
18		vex 3426	. . . . . . . . . . . . . . . . 17 ⊢ 𝑧 ∈ V
19		vex 3426	. . . . . . . . . . . . . . . . 17 ⊢ 𝑤 ∈ V
20	18, 19	op1stb 5380	. . . . . . . . . . . . . . . 16 ⊢ ∩ ∩ ⟨𝑧, 𝑤⟩ = 𝑧
21	17, 20	eqtrdi 2795	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → ∩ ∩ 𝑦 = 𝑧)
22		eqeq1 2742	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ∩ ∩ 𝑦 → (𝑥 = 𝑧 ↔ ∩ ∩ 𝑦 = 𝑧))
23	21, 22	syl5ibr 245	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ∩ ∩ 𝑦 → (𝑦 = ⟨𝑧, 𝑤⟩ → 𝑥 = 𝑧))
24		opeq1 4801	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → ⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩)
25	23, 24	syl6 35	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∩ ∩ 𝑦 → (𝑦 = ⟨𝑧, 𝑤⟩ → ⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩))
26	25	imp 406	. . . . . . . . . . . 12 ⊢ ((𝑥 = ∩ ∩ 𝑦 ∧ 𝑦 = ⟨𝑧, 𝑤⟩) → ⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩)
27		eqeq2 2750	. . . . . . . . . . . . . 14 ⊢ (⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩ → (𝑦 = ⟨𝑥, 𝑤⟩ ↔ 𝑦 = ⟨𝑧, 𝑤⟩))
28	27	biimprcd 249	. . . . . . . . . . . . 13 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → (⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩ → 𝑦 = ⟨𝑥, 𝑤⟩))
29	28	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑥 = ∩ ∩ 𝑦 ∧ 𝑦 = ⟨𝑧, 𝑤⟩) → (⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩ → 𝑦 = ⟨𝑥, 𝑤⟩))
30	26, 29	mpd 15	. . . . . . . . . . 11 ⊢ ((𝑥 = ∩ ∩ 𝑦 ∧ 𝑦 = ⟨𝑧, 𝑤⟩) → 𝑦 = ⟨𝑥, 𝑤⟩)
31	30	ancoms 458	. . . . . . . . . 10 ⊢ ((𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = ∩ ∩ 𝑦) → 𝑦 = ⟨𝑥, 𝑤⟩)
32	31	adantl 481	. . . . . . . . 9 ⊢ (((Fun 𝐹 ∧ 𝑦 ∈ 𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = ∩ ∩ 𝑦)) → 𝑦 = ⟨𝑥, 𝑤⟩)
33	30	eleq1d 2823	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = ∩ ∩ 𝑦 ∧ 𝑦 = ⟨𝑧, 𝑤⟩) → (𝑦 ∈ 𝐹 ↔ ⟨𝑥, 𝑤⟩ ∈ 𝐹))
34	33	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐹 ∧ (𝑥 = ∩ ∩ 𝑦 ∧ 𝑦 = ⟨𝑧, 𝑤⟩)) → (𝑦 ∈ 𝐹 ↔ ⟨𝑥, 𝑤⟩ ∈ 𝐹))
35		funopfv 6803	. . . . . . . . . . . . . . 15 ⊢ (Fun 𝐹 → (⟨𝑥, 𝑤⟩ ∈ 𝐹 → (𝐹‘𝑥) = 𝑤))
36	35	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐹 ∧ (𝑥 = ∩ ∩ 𝑦 ∧ 𝑦 = ⟨𝑧, 𝑤⟩)) → (⟨𝑥, 𝑤⟩ ∈ 𝐹 → (𝐹‘𝑥) = 𝑤))
37	34, 36	sylbid 239	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐹 ∧ (𝑥 = ∩ ∩ 𝑦 ∧ 𝑦 = ⟨𝑧, 𝑤⟩)) → (𝑦 ∈ 𝐹 → (𝐹‘𝑥) = 𝑤))
38	37	exp32 420	. . . . . . . . . . . 12 ⊢ (Fun 𝐹 → (𝑥 = ∩ ∩ 𝑦 → (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑦 ∈ 𝐹 → (𝐹‘𝑥) = 𝑤))))
39	38	com24 95	. . . . . . . . . . 11 ⊢ (Fun 𝐹 → (𝑦 ∈ 𝐹 → (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑥 = ∩ ∩ 𝑦 → (𝐹‘𝑥) = 𝑤))))
40	39	imp43 427	. . . . . . . . . 10 ⊢ (((Fun 𝐹 ∧ 𝑦 ∈ 𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = ∩ ∩ 𝑦)) → (𝐹‘𝑥) = 𝑤)
41	40	opeq2d 4808	. . . . . . . . 9 ⊢ (((Fun 𝐹 ∧ 𝑦 ∈ 𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = ∩ ∩ 𝑦)) → ⟨𝑥, (𝐹‘𝑥)⟩ = ⟨𝑥, 𝑤⟩)
42	32, 41	eqtr4d 2781	. . . . . . . 8 ⊢ (((Fun 𝐹 ∧ 𝑦 ∈ 𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = ∩ ∩ 𝑦)) → 𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩)
43	42	exp32 420	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑥 = ∩ ∩ 𝑦 → 𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩)))
44	43	exlimdvv 1938	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ 𝐹) → (∃𝑧∃𝑤 𝑦 = ⟨𝑧, 𝑤⟩ → (𝑥 = ∩ ∩ 𝑦 → 𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩)))
45	15, 44	mpd 15	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 = ∩ ∩ 𝑦 → 𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩))
46	45	adantrl 712	. . . 4 ⊢ ((Fun 𝐹 ∧ (𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝐹)) → (𝑥 = ∩ ∩ 𝑦 → 𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩))
47		inteq 4879	. . . . . 6 ⊢ (𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩ → ∩ 𝑦 = ∩ ⟨𝑥, (𝐹‘𝑥)⟩)
48	47	inteqd 4881	. . . . 5 ⊢ (𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩ → ∩ ∩ 𝑦 = ∩ ∩ ⟨𝑥, (𝐹‘𝑥)⟩)
49		vex 3426	. . . . . 6 ⊢ 𝑥 ∈ V
50		fvex 6769	. . . . . 6 ⊢ (𝐹‘𝑥) ∈ V
51	49, 50	op1stb 5380	. . . . 5 ⊢ ∩ ∩ ⟨𝑥, (𝐹‘𝑥)⟩ = 𝑥
52	48, 51	eqtr2di 2796	. . . 4 ⊢ (𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩ → 𝑥 = ∩ ∩ 𝑦)
53	46, 52	impbid1 224	. . 3 ⊢ ((Fun 𝐹 ∧ (𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝐹)) → (𝑥 = ∩ ∩ 𝑦 ↔ 𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩))
54	53	ex 412	. 2 ⊢ (Fun 𝐹 → ((𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 = ∩ ∩ 𝑦 ↔ 𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩)))
55	3, 4, 6, 10, 54	en3d 8732	1 ⊢ (Fun 𝐹 → dom 𝐹 ≈ 𝐹)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  Vcvv 3422   ⊆ wss 3883  ⟨cop 4564  ∩ cint 4876   class class class wbr 5070   × cxp 5578  dom cdm 5580  Rel wrel 5585  Fun wfun 6412  ‘cfv 6418   ≈ cen 8688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-int 4877  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-en 8692

This theorem is referenced by:  fundmeng  8776  infmap2  9905  heicant  35739