Theorem fnsex 5833

Description: The function with domain relationship exists. (Contributed by SF, 23-Feb-2015.)

Assertion

Ref	Expression
fnsex	⊢ Fns ∈ V

Proof of Theorem fnsex

Dummy variables f a are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fns 5763	. . 3 ⊢ Fns = {⟨f, a⟩ ∣ f Fn a}
2		vex 2863	. . . . . . . 8 ⊢ a ∈ V
3		opelxp 4812	. . . . . . . 8 ⊢ (⟨f, a⟩ ∈ ( Funs × V) ↔ (f ∈ Funs ∧ a ∈ V))
4	2, 3	mpbiran2 885	. . . . . . 7 ⊢ (⟨f, a⟩ ∈ ( Funs × V) ↔ f ∈ Funs )
5		vex 2863	. . . . . . . 8 ⊢ f ∈ V
6	5	elfuns 5830	. . . . . . 7 ⊢ (f ∈ Funs ↔ Fun f)
7	4, 6	bitri 240	. . . . . 6 ⊢ (⟨f, a⟩ ∈ ( Funs × V) ↔ Fun f)
8		eqcom 2355	. . . . . . 7 ⊢ ((1st “ f) = a ↔ a = (1st “ f))
9		dfdm4 5508	. . . . . . . 8 ⊢ dom f = (1st “ f)
10	9	eqeq1i 2360	. . . . . . 7 ⊢ (dom f = a ↔ (1st “ f) = a)
11		df-br 4641	. . . . . . . 8 ⊢ (fImage1st a ↔ ⟨f, a⟩ ∈ Image1st )
12	5, 2	brimage 5794	. . . . . . . 8 ⊢ (fImage1st a ↔ a = (1st “ f))
13	11, 12	bitr3i 242	. . . . . . 7 ⊢ (⟨f, a⟩ ∈ Image1st ↔ a = (1st “ f))
14	8, 10, 13	3bitr4ri 269	. . . . . 6 ⊢ (⟨f, a⟩ ∈ Image1st ↔ dom f = a)
15	7, 14	anbi12i 678	. . . . 5 ⊢ ((⟨f, a⟩ ∈ ( Funs × V) ∧ ⟨f, a⟩ ∈ Image1st ) ↔ (Fun f ∧ dom f = a))
16		elin 3220	. . . . 5 ⊢ (⟨f, a⟩ ∈ (( Funs × V) ∩ Image1st ) ↔ (⟨f, a⟩ ∈ ( Funs × V) ∧ ⟨f, a⟩ ∈ Image1st ))
17		df-fn 4791	. . . . 5 ⊢ (f Fn a ↔ (Fun f ∧ dom f = a))
18	15, 16, 17	3bitr4i 268	. . . 4 ⊢ (⟨f, a⟩ ∈ (( Funs × V) ∩ Image1st ) ↔ f Fn a)
19	18	opabbi2i 4867	. . 3 ⊢ (( Funs × V) ∩ Image1st ) = {⟨f, a⟩ ∣ f Fn a}
20	1, 19	eqtr4i 2376	. 2 ⊢ Fns = (( Funs × V) ∩ Image1st )
21		funsex 5829	. . . 4 ⊢ Funs ∈ V
22		vvex 4110	. . . 4 ⊢ V ∈ V
23	21, 22	xpex 5116	. . 3 ⊢ ( Funs × V) ∈ V
24		1stex 4740	. . . 4 ⊢ 1st ∈ V
25	24	imageex 5802	. . 3 ⊢ Image1st ∈ V
26	23, 25	inex 4106	. 2 ⊢ (( Funs × V) ∩ Image1st ) ∈ V
27	20, 26	eqeltri 2423	1 ⊢ Fns ∈ V

Syntax hints:   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2860   ∩ cin 3209  ⟨cop 4562  {copab 4623   class class class wbr 4640  1^st c1st 4718   “ cima 4723   × cxp 4771  dom cdm 4773  Fun wfun 4776   Fn wfn 4777  Imagecimage 5754   Funs cfuns 5760   Fns cfns 5762

This theorem was proved from 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 theorem 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-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-2nd 4798  df-txp 5737  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763

This theorem is referenced by:  enex  6032  ovcelem1  6172  ceex  6175