Theorem fundmen 6044

Description: A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98. (Contributed by SF, 23-Feb-2015.)

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 a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssv 3292	. . . . . 6 ⊢ F ⊆ V
2		1stfo 5506	. . . . . . 7 ⊢ 1st :V–onto→V
3		fofn 5272	. . . . . . 7 ⊢ (1st :V–onto→V → 1st Fn V)
4		fnssresb 5196	. . . . . . 7 ⊢ (1st Fn V → ((1st ↾ F) Fn F ↔ F ⊆ V))
5	2, 3, 4	mp2b 9	. . . . . 6 ⊢ ((1st ↾ F) Fn F ↔ F ⊆ V)
6	1, 5	mpbir 200	. . . . 5 ⊢ (1st ↾ F) Fn F
7	6	a1i 10	. . . 4 ⊢ (Fun F → (1st ↾ F) Fn F)
8		brcnv 4893	. . . . . . . . . . 11 ⊢ (x◡(1st ↾ F)y ↔ y(1st ↾ F)x)
9		brres 4950	. . . . . . . . . . 11 ⊢ (y(1st ↾ F)x ↔ (y1st x ∧ y ∈ F))
10		vex 2863	. . . . . . . . . . . . . 14 ⊢ x ∈ V
11	10	br1st 4859	. . . . . . . . . . . . 13 ⊢ (y1st x ↔ ∃a y = ⟨x, a⟩)
12	11	anbi1i 676	. . . . . . . . . . . 12 ⊢ ((y1st x ∧ y ∈ F) ↔ (∃a y = ⟨x, a⟩ ∧ y ∈ F))
13		19.41v 1901	. . . . . . . . . . . 12 ⊢ (∃a(y = ⟨x, a⟩ ∧ y ∈ F) ↔ (∃a y = ⟨x, a⟩ ∧ y ∈ F))
14	12, 13	bitr4i 243	. . . . . . . . . . 11 ⊢ ((y1st x ∧ y ∈ F) ↔ ∃a(y = ⟨x, a⟩ ∧ y ∈ F))
15	8, 9, 14	3bitri 262	. . . . . . . . . 10 ⊢ (x◡(1st ↾ F)y ↔ ∃a(y = ⟨x, a⟩ ∧ y ∈ F))
16		brcnv 4893	. . . . . . . . . . . 12 ⊢ (x◡(1st ↾ F)z ↔ z(1st ↾ F)x)
17		brres 4950	. . . . . . . . . . . 12 ⊢ (z(1st ↾ F)x ↔ (z1st x ∧ z ∈ F))
18	10	br1st 4859	. . . . . . . . . . . . 13 ⊢ (z1st x ↔ ∃b z = ⟨x, b⟩)
19	18	anbi1i 676	. . . . . . . . . . . 12 ⊢ ((z1st x ∧ z ∈ F) ↔ (∃b z = ⟨x, b⟩ ∧ z ∈ F))
20	16, 17, 19	3bitri 262	. . . . . . . . . . 11 ⊢ (x◡(1st ↾ F)z ↔ (∃b z = ⟨x, b⟩ ∧ z ∈ F))
21		19.41v 1901	. . . . . . . . . . 11 ⊢ (∃b(z = ⟨x, b⟩ ∧ z ∈ F) ↔ (∃b z = ⟨x, b⟩ ∧ z ∈ F))
22	20, 21	bitr4i 243	. . . . . . . . . 10 ⊢ (x◡(1st ↾ F)z ↔ ∃b(z = ⟨x, b⟩ ∧ z ∈ F))
23	15, 22	anbi12i 678	. . . . . . . . 9 ⊢ ((x◡(1st ↾ F)y ∧ x◡(1st ↾ F)z) ↔ (∃a(y = ⟨x, a⟩ ∧ y ∈ F) ∧ ∃b(z = ⟨x, b⟩ ∧ z ∈ F)))
24		eeanv 1913	. . . . . . . . 9 ⊢ (∃a∃b((y = ⟨x, a⟩ ∧ y ∈ F) ∧ (z = ⟨x, b⟩ ∧ z ∈ F)) ↔ (∃a(y = ⟨x, a⟩ ∧ y ∈ F) ∧ ∃b(z = ⟨x, b⟩ ∧ z ∈ F)))
25	23, 24	bitr4i 243	. . . . . . . 8 ⊢ ((x◡(1st ↾ F)y ∧ x◡(1st ↾ F)z) ↔ ∃a∃b((y = ⟨x, a⟩ ∧ y ∈ F) ∧ (z = ⟨x, b⟩ ∧ z ∈ F)))
26		an4 797	. . . . . . . . . 10 ⊢ (((y = ⟨x, a⟩ ∧ y ∈ F) ∧ (z = ⟨x, b⟩ ∧ z ∈ F)) ↔ ((y = ⟨x, a⟩ ∧ z = ⟨x, b⟩) ∧ (y ∈ F ∧ z ∈ F)))
27		dffun4 5122	. . . . . . . . . . . . 13 ⊢ (Fun F ↔ ∀x∀a∀b((⟨x, a⟩ ∈ F ∧ ⟨x, b⟩ ∈ F) → a = b))
28		sp 1747	. . . . . . . . . . . . . . 15 ⊢ (∀b((⟨x, a⟩ ∈ F ∧ ⟨x, b⟩ ∈ F) → a = b) → ((⟨x, a⟩ ∈ F ∧ ⟨x, b⟩ ∈ F) → a = b))
29	28	sps 1754	. . . . . . . . . . . . . 14 ⊢ (∀a∀b((⟨x, a⟩ ∈ F ∧ ⟨x, b⟩ ∈ F) → a = b) → ((⟨x, a⟩ ∈ F ∧ ⟨x, b⟩ ∈ F) → a = b))
30	29	sps 1754	. . . . . . . . . . . . 13 ⊢ (∀x∀a∀b((⟨x, a⟩ ∈ F ∧ ⟨x, b⟩ ∈ F) → a = b) → ((⟨x, a⟩ ∈ F ∧ ⟨x, b⟩ ∈ F) → a = b))
31	27, 30	sylbi 187	. . . . . . . . . . . 12 ⊢ (Fun F → ((⟨x, a⟩ ∈ F ∧ ⟨x, b⟩ ∈ F) → a = b))
32		opeq2 4580	. . . . . . . . . . . 12 ⊢ (a = b → ⟨x, a⟩ = ⟨x, b⟩)
33	31, 32	syl6 29	. . . . . . . . . . 11 ⊢ (Fun F → ((⟨x, a⟩ ∈ F ∧ ⟨x, b⟩ ∈ F) → ⟨x, a⟩ = ⟨x, b⟩))
34		eleq1 2413	. . . . . . . . . . . . . . 15 ⊢ (y = ⟨x, a⟩ → (y ∈ F ↔ ⟨x, a⟩ ∈ F))
35		eleq1 2413	. . . . . . . . . . . . . . 15 ⊢ (z = ⟨x, b⟩ → (z ∈ F ↔ ⟨x, b⟩ ∈ F))
36	34, 35	bi2anan9 843	. . . . . . . . . . . . . 14 ⊢ ((y = ⟨x, a⟩ ∧ z = ⟨x, b⟩) → ((y ∈ F ∧ z ∈ F) ↔ (⟨x, a⟩ ∈ F ∧ ⟨x, b⟩ ∈ F)))
37		eqeq12 2365	. . . . . . . . . . . . . 14 ⊢ ((y = ⟨x, a⟩ ∧ z = ⟨x, b⟩) → (y = z ↔ ⟨x, a⟩ = ⟨x, b⟩))
38	36, 37	imbi12d 311	. . . . . . . . . . . . 13 ⊢ ((y = ⟨x, a⟩ ∧ z = ⟨x, b⟩) → (((y ∈ F ∧ z ∈ F) → y = z) ↔ ((⟨x, a⟩ ∈ F ∧ ⟨x, b⟩ ∈ F) → ⟨x, a⟩ = ⟨x, b⟩)))
39	38	biimprcd 216	. . . . . . . . . . . 12 ⊢ (((⟨x, a⟩ ∈ F ∧ ⟨x, b⟩ ∈ F) → ⟨x, a⟩ = ⟨x, b⟩) → ((y = ⟨x, a⟩ ∧ z = ⟨x, b⟩) → ((y ∈ F ∧ z ∈ F) → y = z)))
40	39	imp3a 420	. . . . . . . . . . 11 ⊢ (((⟨x, a⟩ ∈ F ∧ ⟨x, b⟩ ∈ F) → ⟨x, a⟩ = ⟨x, b⟩) → (((y = ⟨x, a⟩ ∧ z = ⟨x, b⟩) ∧ (y ∈ F ∧ z ∈ F)) → y = z))
41	33, 40	syl 15	. . . . . . . . . 10 ⊢ (Fun F → (((y = ⟨x, a⟩ ∧ z = ⟨x, b⟩) ∧ (y ∈ F ∧ z ∈ F)) → y = z))
42	26, 41	syl5bi 208	. . . . . . . . 9 ⊢ (Fun F → (((y = ⟨x, a⟩ ∧ y ∈ F) ∧ (z = ⟨x, b⟩ ∧ z ∈ F)) → y = z))
43	42	exlimdvv 1637	. . . . . . . 8 ⊢ (Fun F → (∃a∃b((y = ⟨x, a⟩ ∧ y ∈ F) ∧ (z = ⟨x, b⟩ ∧ z ∈ F)) → y = z))
44	25, 43	syl5bi 208	. . . . . . 7 ⊢ (Fun F → ((x◡(1st ↾ F)y ∧ x◡(1st ↾ F)z) → y = z))
45	44	alrimiv 1631	. . . . . 6 ⊢ (Fun F → ∀z((x◡(1st ↾ F)y ∧ x◡(1st ↾ F)z) → y = z))
46	45	alrimivv 1632	. . . . 5 ⊢ (Fun F → ∀x∀y∀z((x◡(1st ↾ F)y ∧ x◡(1st ↾ F)z) → y = z))
47		dffun2 5120	. . . . 5 ⊢ (Fun ◡(1st ↾ F) ↔ ∀x∀y∀z((x◡(1st ↾ F)y ∧ x◡(1st ↾ F)z) → y = z))
48	46, 47	sylibr 203	. . . 4 ⊢ (Fun F → Fun ◡(1st ↾ F))
49		dfdm4 5508	. . . . . 6 ⊢ dom F = (1st “ F)
50		dfima3 4952	. . . . . 6 ⊢ (1st “ F) = ran (1st ↾ F)
51	49, 50	eqtr2i 2374	. . . . 5 ⊢ ran (1st ↾ F) = dom F
52	51	a1i 10	. . . 4 ⊢ (Fun F → ran (1st ↾ F) = dom F)
53		dff1o2 5292	. . . 4 ⊢ ((1st ↾ F):F–1-1-onto→dom F ↔ ((1st ↾ F) Fn F ∧ Fun ◡(1st ↾ F) ∧ ran (1st ↾ F) = dom F))
54	7, 48, 52, 53	syl3anbrc 1136	. . 3 ⊢ (Fun F → (1st ↾ F):F–1-1-onto→dom F)
55		1stex 4740	. . . . 5 ⊢ 1st ∈ V
56		fundmen.1	. . . . 5 ⊢ F ∈ V
57	55, 56	resex 5118	. . . 4 ⊢ (1st ↾ F) ∈ V
58	57	f1oen 6034	. . 3 ⊢ ((1st ↾ F):F–1-1-onto→dom F → F ≈ dom F)
59	54, 58	syl 15	. 2 ⊢ (Fun F → F ≈ dom F)
60		ensym 6038	. 2 ⊢ (F ≈ dom F ↔ dom F ≈ F)
61	59, 60	sylib 188	1 ⊢ (Fun F → dom F ≈ F)

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860   ⊆ wss 3258  ⟨cop 4562   class class class wbr 4640  1^st c1st 4718   “ cima 4723  ^◡ccnv 4772  dom cdm 4773  ran crn 4774   ↾ cres 4775  Fun wfun 4776   Fn wfn 4777  –onto→wfo 4780  –1-1-onto→wf1o 4781   ≈ cen 6029

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-co 4727  df-ima 4728  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-2nd 4798  df-en 6030

This theorem is used by:  fundmeng  6045  xpsnen  6050