Theorem fundmen 3333

Description: A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98.

Hypothesis

Ref	Expression
fundmen.1	⊢ F ∈ V

Assertion

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

Proof of Theorem fundmen

Step	Hyp	Ref	Expression
1		fundmen.1	. . . 4 ⊢ F ∈ V
2		dmexg 2551	. . . 4 ⊢ (F ∈ V → dom F ∈ V)
3	1, 2	ax-mp 6	. . 3 ⊢ dom F ∈ V
4	3	a1i 7	. 2 ⊢ (Fun F → dom F ∈ V)
5		funopfv 2886	. . 3 ⊢ ((Fun F ∧ x ∈ dom F) → ⟨x, (F ‘x)⟩ ∈ F)
6	5	exp 291	. 2 ⊢ (Fun F → (x ∈ dom F → ⟨x, (F ‘x)⟩ ∈ F))
7		funrel 2681	. . 3 ⊢ (Fun F → Rel F)
8		elreldm 2554	. . . 4 ⊢ ((Rel F ∧ y ∈ F) → ∩∩y ∈ dom F)
9	8	exp 291	. . 3 ⊢ (Rel F → (y ∈ F → ∩∩y ∈ dom F))
10	7, 9	syl 12	. 2 ⊢ (Fun F → (y ∈ F → ∩∩y ∈ dom F))
11		ssel2 1503	. . . . . . . 8 ⊢ ((F ⊆ (V × V) ∧ y ∈ F) → y ∈ (V × V))
12		df-rel 2425	. . . . . . . . 9 ⊢ (Rel F ↔ F ⊆ (V × V))
13	7, 12	sylib 173	. . . . . . . 8 ⊢ (Fun F → F ⊆ (V × V))
14	11, 13	sylan 343	. . . . . . 7 ⊢ ((Fun F ∧ y ∈ F) → y ∈ (V × V))
15		elvv 2464	. . . . . . 7 ⊢ (y ∈ (V × V) ↔ ∃z∃w y = ⟨z, w⟩)
16	14, 15	sylib 173	. . . . . 6 ⊢ ((Fun F ∧ y ∈ F) → ∃z∃w y = ⟨z, w⟩)
17		cleq1 1107	. . . . . . . . . . . . . . 15 ⊢ (x = ∩∩y → (x = z ↔ ∩∩y = z))
18		inteq 1968	. . . . . . . . . . . . . . . . 17 ⊢ (y = ⟨z, w⟩ → ∩y = ∩⟨z, w⟩)
19	18	inteqd 1970	. . . . . . . . . . . . . . . 16 ⊢ (y = ⟨z, w⟩ → ∩∩y = ∩∩⟨z, w⟩)
20		visset 1350	. . . . . . . . . . . . . . . . 17 ⊢ z ∈ V
21	20	op1stb 1992	. . . . . . . . . . . . . . . 16 ⊢ ∩∩⟨z, w⟩ = z
22	19, 21	syl6eq 1140	. . . . . . . . . . . . . . 15 ⊢ (y = ⟨z, w⟩ → ∩∩y = z)
23	17, 22	syl5bir 184	. . . . . . . . . . . . . 14 ⊢ (x = ∩∩y → (y = ⟨z, w⟩ → x = z))
24		opeq1 1876	. . . . . . . . . . . . . 14 ⊢ (x = z → ⟨x, w⟩ = ⟨z, w⟩)
25	23, 24	syl6 23	. . . . . . . . . . . . 13 ⊢ (x = ∩∩y → (y = ⟨z, w⟩ → ⟨x, w⟩ = ⟨z, w⟩))
26	25	imp 277	. . . . . . . . . . . 12 ⊢ ((x = ∩∩y ∧ y = ⟨z, w⟩) → ⟨x, w⟩ = ⟨z, w⟩)
27		cleq2 1110	. . . . . . . . . . . . . 14 ⊢ (⟨x, w⟩ = ⟨z, w⟩ → (y = ⟨x, w⟩ ↔ y = ⟨z, w⟩))
28	27	biimprcd 138	. . . . . . . . . . . . 13 ⊢ (y = ⟨z, w⟩ → (⟨x, w⟩ = ⟨z, w⟩ → y = ⟨x, w⟩))
29	28	adantl 305	. . . . . . . . . . . 12 ⊢ ((x = ∩∩y ∧ y = ⟨z, w⟩) → (⟨x, w⟩ = ⟨z, w⟩ → y = ⟨x, w⟩))
30	26, 29	mpd 46	. . . . . . . . . . 11 ⊢ ((x = ∩∩y ∧ y = ⟨z, w⟩) → y = ⟨x, w⟩)
31	30	ancoms 334	. . . . . . . . . 10 ⊢ ((y = ⟨z, w⟩ ∧ x = ∩∩y) → y = ⟨x, w⟩)
32	31	adantl 305	. . . . . . . . 9 ⊢ (((Fun F ∧ y ∈ F) ∧ (y = ⟨z, w⟩ ∧ x = ∩∩y)) → y = ⟨x, w⟩)
33	30	eleq1d 1155	. . . . . . . . . . . . . . 15 ⊢ ((x = ∩∩y ∧ y = ⟨z, w⟩) → (y ∈ F ↔ ⟨x, w⟩ ∈ F))
34	33	adantl 305	. . . . . . . . . . . . . 14 ⊢ ((Fun F ∧ (x = ∩∩y ∧ y = ⟨z, w⟩)) → (y ∈ F ↔ ⟨x, w⟩ ∈ F))
35		visset 1350	. . . . . . . . . . . . . . . 16 ⊢ w ∈ V
36	35	funfvopi 2853	. . . . . . . . . . . . . . 15 ⊢ (Fun F → (⟨x, w⟩ ∈ F → (F ‘x) = w))
37	36	adantr 306	. . . . . . . . . . . . . 14 ⊢ ((Fun F ∧ (x = ∩∩y ∧ y = ⟨z, w⟩)) → (⟨x, w⟩ ∈ F → (F ‘x) = w))
38	34, 37	sylbid 178	. . . . . . . . . . . . 13 ⊢ ((Fun F ∧ (x = ∩∩y ∧ y = ⟨z, w⟩)) → (y ∈ F → (F ‘x) = w))
39	38	exp32 294	. . . . . . . . . . . 12 ⊢ (Fun F → (x = ∩∩y → (y = ⟨z, w⟩ → (y ∈ F → (F ‘x) = w))))
40	39	com24 37	. . . . . . . . . . 11 ⊢ (Fun F → (y ∈ F → (y = ⟨z, w⟩ → (x = ∩∩y → (F ‘x) = w))))
41	40	imp43 288	. . . . . . . . . 10 ⊢ (((Fun F ∧ y ∈ F) ∧ (y = ⟨z, w⟩ ∧ x = ∩∩y)) → (F ‘x) = w)
42		opeq2 1877	. . . . . . . . . 10 ⊢ ((F ‘x) = w → ⟨x, (F ‘x)⟩ = ⟨x, w⟩)
43	41, 42	syl 12	. . . . . . . . 9 ⊢ (((Fun F ∧ y ∈ F) ∧ (y = ⟨z, w⟩ ∧ x = ∩∩y)) → ⟨x, (F ‘x)⟩ = ⟨x, w⟩)
44	32, 43	eqtr4d 1131	. . . . . . . 8 ⊢ (((Fun F ∧ y ∈ F) ∧ (y = ⟨z, w⟩ ∧ x = ∩∩y)) → y = ⟨x, (F ‘x)⟩)
45	44	exp32 294	. . . . . . 7 ⊢ ((Fun F ∧ y ∈ F) → (y = ⟨z, w⟩ → (x = ∩∩y → y = ⟨x, (F ‘x)⟩)))
46	45	19.23advv 955	. . . . . 6 ⊢ ((Fun F ∧ y ∈ F) → (∃z∃w y = ⟨z, w⟩ → (x = ∩∩y → y = ⟨x, (F ‘x)⟩)))
47	16, 46	mpd 46	. . . . 5 ⊢ ((Fun F ∧ y ∈ F) → (x = ∩∩y → y = ⟨x, (F ‘x)⟩))
48	47	adantrl 311	. . . 4 ⊢ ((Fun F ∧ (x ∈ dom F ∧ y ∈ F)) → (x = ∩∩y → y = ⟨x, (F ‘x)⟩))
49		inteq 1968	. . . . . . 7 ⊢ (y = ⟨x, (F ‘x)⟩ → ∩y = ∩⟨x, (F ‘x)⟩)
50	49	inteqd 1970	. . . . . 6 ⊢ (y = ⟨x, (F ‘x)⟩ → ∩∩y = ∩∩⟨x, (F ‘x)⟩)
51		visset 1350	. . . . . . 7 ⊢ x ∈ V
52	51	op1stb 1992	. . . . . 6 ⊢ ∩∩⟨x, (F ‘x)⟩ = x
53	50, 52	syl6req 1141	. . . . 5 ⊢ (y = ⟨x, (F ‘x)⟩ → x = ∩∩y)
54	53	a1i 7	. . . 4 ⊢ ((Fun F ∧ (x ∈ dom F ∧ y ∈ F)) → (y = ⟨x, (F ‘x)⟩ → x = ∩∩y))
55	48, 54	impbid 397	. . 3 ⊢ ((Fun F ∧ (x ∈ dom F ∧ y ∈ F)) → (x = ∩∩y ↔ y = ⟨x, (F ‘x)⟩))
56	55	exp 291	. 2 ⊢ (Fun F → ((x ∈ dom F ∧ y ∈ F) → (x = ∩∩y ↔ y = ⟨x, (F ‘x)⟩)))
57	4, 6, 10, 56	en3d 3304	1 ⊢ (Fun F → dom F ≈ F)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678   = weq 797   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487  ⟨cop 1810  ∩cint 1965   class class class wbr 2054   × cxp 2408  dom cdm 2410  Rel wrel 2415  Fun wfun 2416   ‘cfv 2422   ≈ cen 3271

This theorem is referenced by:  infmap2 4953

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-int 1966  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-en 3274

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