Theorem fundmen 8030

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 7099	. . 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 6329	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹)
6	5	ex 450	. 2 ⊢ (Fun 𝐹 → (𝑥 ∈ dom 𝐹 → ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹))
7		funrel 5905	. . 3 ⊢ (Fun 𝐹 → Rel 𝐹)
8		elreldm 5350	. . . 4 ⊢ ((Rel 𝐹 ∧ 𝑦 ∈ 𝐹) → ∩ ∩ 𝑦 ∈ dom 𝐹)
9	8	ex 450	. . 3 ⊢ (Rel 𝐹 → (𝑦 ∈ 𝐹 → ∩ ∩ 𝑦 ∈ dom 𝐹))
10	7, 9	syl 17	. 2 ⊢ (Fun 𝐹 → (𝑦 ∈ 𝐹 → ∩ ∩ 𝑦 ∈ dom 𝐹))
11		df-rel 5121	. . . . . . . . 9 ⊢ (Rel 𝐹 ↔ 𝐹 ⊆ (V × V))
12	7, 11	sylib 208	. . . . . . . 8 ⊢ (Fun 𝐹 → 𝐹 ⊆ (V × V))
13	12	sselda 3603	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ 𝐹) → 𝑦 ∈ (V × V))
14		elvv 5177	. . . . . . 7 ⊢ (𝑦 ∈ (V × V) ↔ ∃𝑧∃𝑤 𝑦 = ⟨𝑧, 𝑤⟩)
15	13, 14	sylib 208	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ 𝐹) → ∃𝑧∃𝑤 𝑦 = ⟨𝑧, 𝑤⟩)
16		inteq 4478	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → ∩ 𝑦 = ∩ ⟨𝑧, 𝑤⟩)
17	16	inteqd 4480	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → ∩ ∩ 𝑦 = ∩ ∩ ⟨𝑧, 𝑤⟩)
18		vex 3203	. . . . . . . . . . . . . . . . 17 ⊢ 𝑧 ∈ V
19		vex 3203	. . . . . . . . . . . . . . . . 17 ⊢ 𝑤 ∈ V
20	18, 19	op1stb 4940	. . . . . . . . . . . . . . . 16 ⊢ ∩ ∩ ⟨𝑧, 𝑤⟩ = 𝑧
21	17, 20	syl6eq 2672	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → ∩ ∩ 𝑦 = 𝑧)
22		eqeq1 2626	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ∩ ∩ 𝑦 → (𝑥 = 𝑧 ↔ ∩ ∩ 𝑦 = 𝑧))
23	21, 22	syl5ibr 236	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ∩ ∩ 𝑦 → (𝑦 = ⟨𝑧, 𝑤⟩ → 𝑥 = 𝑧))
24		opeq1 4402	. . . . . . . . . . . . . 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 445	. . . . . . . . . . . 12 ⊢ ((𝑥 = ∩ ∩ 𝑦 ∧ 𝑦 = ⟨𝑧, 𝑤⟩) → ⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩)
27		eqeq2 2633	. . . . . . . . . . . . . 14 ⊢ (⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩ → (𝑦 = ⟨𝑥, 𝑤⟩ ↔ 𝑦 = ⟨𝑧, 𝑤⟩))
28	27	biimprcd 240	. . . . . . . . . . . . 13 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → (⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩ → 𝑦 = ⟨𝑥, 𝑤⟩))
29	28	adantl 482	. . . . . . . . . . . 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 469	. . . . . . . . . 10 ⊢ ((𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = ∩ ∩ 𝑦) → 𝑦 = ⟨𝑥, 𝑤⟩)
32	31	adantl 482	. . . . . . . . 9 ⊢ (((Fun 𝐹 ∧ 𝑦 ∈ 𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = ∩ ∩ 𝑦)) → 𝑦 = ⟨𝑥, 𝑤⟩)
33	30	eleq1d 2686	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = ∩ ∩ 𝑦 ∧ 𝑦 = ⟨𝑧, 𝑤⟩) → (𝑦 ∈ 𝐹 ↔ ⟨𝑥, 𝑤⟩ ∈ 𝐹))
34	33	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐹 ∧ (𝑥 = ∩ ∩ 𝑦 ∧ 𝑦 = ⟨𝑧, 𝑤⟩)) → (𝑦 ∈ 𝐹 ↔ ⟨𝑥, 𝑤⟩ ∈ 𝐹))
35		funopfv 6235	. . . . . . . . . . . . . . 15 ⊢ (Fun 𝐹 → (⟨𝑥, 𝑤⟩ ∈ 𝐹 → (𝐹‘𝑥) = 𝑤))
36	35	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐹 ∧ (𝑥 = ∩ ∩ 𝑦 ∧ 𝑦 = ⟨𝑧, 𝑤⟩)) → (⟨𝑥, 𝑤⟩ ∈ 𝐹 → (𝐹‘𝑥) = 𝑤))
37	34, 36	sylbid 230	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐹 ∧ (𝑥 = ∩ ∩ 𝑦 ∧ 𝑦 = ⟨𝑧, 𝑤⟩)) → (𝑦 ∈ 𝐹 → (𝐹‘𝑥) = 𝑤))
38	37	exp32 631	. . . . . . . . . . . 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 621	. . . . . . . . . 10 ⊢ (((Fun 𝐹 ∧ 𝑦 ∈ 𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = ∩ ∩ 𝑦)) → (𝐹‘𝑥) = 𝑤)
41	40	opeq2d 4409	. . . . . . . . 9 ⊢ (((Fun 𝐹 ∧ 𝑦 ∈ 𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = ∩ ∩ 𝑦)) → ⟨𝑥, (𝐹‘𝑥)⟩ = ⟨𝑥, 𝑤⟩)
42	32, 41	eqtr4d 2659	. . . . . . . 8 ⊢ (((Fun 𝐹 ∧ 𝑦 ∈ 𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = ∩ ∩ 𝑦)) → 𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩)
43	42	exp32 631	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑥 = ∩ ∩ 𝑦 → 𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩)))
44	43	exlimdvv 1862	. . . . . 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 752	. . . 4 ⊢ ((Fun 𝐹 ∧ (𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝐹)) → (𝑥 = ∩ ∩ 𝑦 → 𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩))
47		inteq 4478	. . . . . 6 ⊢ (𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩ → ∩ 𝑦 = ∩ ⟨𝑥, (𝐹‘𝑥)⟩)
48	47	inteqd 4480	. . . . 5 ⊢ (𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩ → ∩ ∩ 𝑦 = ∩ ∩ ⟨𝑥, (𝐹‘𝑥)⟩)
49		vex 3203	. . . . . 6 ⊢ 𝑥 ∈ V
50		fvex 6201	. . . . . 6 ⊢ (𝐹‘𝑥) ∈ V
51	49, 50	op1stb 4940	. . . . 5 ⊢ ∩ ∩ ⟨𝑥, (𝐹‘𝑥)⟩ = 𝑥
52	48, 51	syl6req 2673	. . . 4 ⊢ (𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩ → 𝑥 = ∩ ∩ 𝑦)
53	46, 52	impbid1 215	. . 3 ⊢ ((Fun 𝐹 ∧ (𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝐹)) → (𝑥 = ∩ ∩ 𝑦 ↔ 𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩))
54	53	ex 450	. 2 ⊢ (Fun 𝐹 → ((𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 = ∩ ∩ 𝑦 ↔ 𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩)))
55	3, 4, 6, 10, 54	en3d 7992	1 ⊢ (Fun 𝐹 → dom 𝐹 ≈ 𝐹)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  Vcvv 3200   ⊆ wss 3574  ⟨cop 4183  ∩ cint 4475   class class class wbr 4653   × cxp 5112  dom cdm 5114  Rel wrel 5119  Fun wfun 5882  ‘cfv 5888   ≈ cen 7952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-en 7956

This theorem is referenced by:  fundmeng  8031  infmap2  9040  heicant  33444