Theorem fundmen 9024

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 7902	. . 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 7045	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹)
6	5	ex 417	. 2 ⊢ (Fun 𝐹 → (𝑥 ∈ dom 𝐹 → ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹))
7		funrel 6553	. . 3 ⊢ (Fun 𝐹 → Rel 𝐹)
8		elreldm 5925	. . . 4 ⊢ ((Rel 𝐹 ∧ 𝑦 ∈ 𝐹) → ∩ ∩ 𝑦 ∈ dom 𝐹)
9	8	ex 417	. . 3 ⊢ (Rel 𝐹 → (𝑦 ∈ 𝐹 → ∩ ∩ 𝑦 ∈ dom 𝐹))
10	7, 9	syl 18	. 2 ⊢ (Fun 𝐹 → (𝑦 ∈ 𝐹 → ∩ ∩ 𝑦 ∈ dom 𝐹))
11		df-rel 5668	. . . . . . . . 9 ⊢ (Rel 𝐹 ↔ 𝐹 ⊆ (V × V))
12	7, 11	sylib 221	. . . . . . . 8 ⊢ (Fun 𝐹 → 𝐹 ⊆ (V × V))
13	12	sselda 3937	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ 𝐹) → 𝑦 ∈ (V × V))
14		elvv 5736	. . . . . . 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 4915	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → ∩ 𝑦 = ∩ ⟨𝑧, 𝑤⟩)
17	16	inteqd 4917	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → ∩ ∩ 𝑦 = ∩ ∩ ⟨𝑧, 𝑤⟩)
18		vex 3459	. . . . . . . . . . . . . . . . 17 ⊢ 𝑧 ∈ V
19		vex 3459	. . . . . . . . . . . . . . . . 17 ⊢ 𝑤 ∈ V
20	18, 19	op1stb 5453	. . . . . . . . . . . . . . . 16 ⊢ ∩ ∩ ⟨𝑧, 𝑤⟩ = 𝑧
21	17, 20	eqtrdi 2814	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → ∩ ∩ 𝑦 = 𝑧)
22		eqeq1 2767	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ∩ ∩ 𝑦 → (𝑥 = 𝑧 ↔ ∩ ∩ 𝑦 = 𝑧))
23	21, 22	imbitrrid 249	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ∩ ∩ 𝑦 → (𝑦 = ⟨𝑧, 𝑤⟩ → 𝑥 = 𝑧))
24		opeq1 4838	. . . . . . . . . . . . . 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 411	. . . . . . . . . . . 12 ⊢ ((𝑥 = ∩ ∩ 𝑦 ∧ 𝑦 = ⟨𝑧, 𝑤⟩) → ⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩)
27		eqeq2 2775	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . 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 463	. . . . . . . . . 10 ⊢ ((𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = ∩ ∩ 𝑦) → 𝑦 = ⟨𝑥, 𝑤⟩)
32	31	adantl 486	. . . . . . . . 9 ⊢ (((Fun 𝐹 ∧ 𝑦 ∈ 𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = ∩ ∩ 𝑦)) → 𝑦 = ⟨𝑥, 𝑤⟩)
33	30	eleq1d 2848	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = ∩ ∩ 𝑦 ∧ 𝑦 = ⟨𝑧, 𝑤⟩) → (𝑦 ∈ 𝐹 ↔ ⟨𝑥, 𝑤⟩ ∈ 𝐹))
34	33	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐹 ∧ (𝑥 = ∩ ∩ 𝑦 ∧ 𝑦 = ⟨𝑧, 𝑤⟩)) → (𝑦 ∈ 𝐹 ↔ ⟨𝑥, 𝑤⟩ ∈ 𝐹))
35		funopfv 6930	. . . . . . . . . . . . . . 15 ⊢ (Fun 𝐹 → (⟨𝑥, 𝑤⟩ ∈ 𝐹 → (𝐹‘𝑥) = 𝑤))
36	35	adantr 485	. . . . . . . . . . . . . 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 425	. . . . . . . . . . . 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 432	. . . . . . . . . 10 ⊢ (((Fun 𝐹 ∧ 𝑦 ∈ 𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = ∩ ∩ 𝑦)) → (𝐹‘𝑥) = 𝑤)
41	40	opeq2d 4845	. . . . . . . . 9 ⊢ (((Fun 𝐹 ∧ 𝑦 ∈ 𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = ∩ ∩ 𝑦)) → ⟨𝑥, (𝐹‘𝑥)⟩ = ⟨𝑥, 𝑤⟩)
42	32, 41	eqtr4d 2801	. . . . . . . 8 ⊢ (((Fun 𝐹 ∧ 𝑦 ∈ 𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = ∩ ∩ 𝑦)) → 𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩)
43	42	exp32 425	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑥 = ∩ ∩ 𝑦 → 𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩)))
44	43	exlimdvv 1964	. . . . . 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 728	. . . 4 ⊢ ((Fun 𝐹 ∧ (𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝐹)) → (𝑥 = ∩ ∩ 𝑦 → 𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩))
47		inteq 4915	. . . . . 6 ⊢ (𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩ → ∩ 𝑦 = ∩ ⟨𝑥, (𝐹‘𝑥)⟩)
48	47	inteqd 4917	. . . . 5 ⊢ (𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩ → ∩ ∩ 𝑦 = ∩ ∩ ⟨𝑥, (𝐹‘𝑥)⟩)
49		vex 3459	. . . . . 6 ⊢ 𝑥 ∈ V
50		fvex 6894	. . . . . 6 ⊢ (𝐹‘𝑥) ∈ V
51	49, 50	op1stb 5453	. . . . 5 ⊢ ∩ ∩ ⟨𝑥, (𝐹‘𝑥)⟩ = 𝑥
52	48, 51	eqtr2di 2815	. . . 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 417	. 2 ⊢ (Fun 𝐹 → ((𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 = ∩ ∩ 𝑦 ↔ 𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩)))
55	3, 4, 6, 10, 54	en3d 8982	1 ⊢ (Fun 𝐹 → dom 𝐹 ≈ 𝐹)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143  Vcvv 3455   ⊆ wss 3905  ⟨cop 4595  ∩ cint 4912   class class class wbr 5109   × cxp 5659  dom cdm 5661  Rel wrel 5666  Fun wfun 6530  ‘cfv 6536   ≈ cen 8936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-en 8940

This theorem is referenced by:  fundmeng  9025  infmap2  10196  heicant  38326