Theorem fnsex 5832

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 5762	. . 3 ⊢ Fns = {⟨f, a⟩ ∣ f Fn a}
2		vex 2862	. . . . . . . 8 ⊢ a ∈ V
3		opelxp 4811	. . . . . . . 8 ⊢ (⟨f, a⟩ ∈ ( Funs × V) ↔ (f ∈ Funs ∧ a ∈ V))
4	2, 3	mpbiran2 885	. . . . . . 7 ⊢ (⟨f, a⟩ ∈ ( Funs × V) ↔ f ∈ Funs )
5		vex 2862	. . . . . . . 8 ⊢ f ∈ V
6	5	elfuns 5829	. . . . . . 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 5507	. . . . . . . 8 ⊢ dom f = (1st “ f)
10	9	eqeq1i 2360	. . . . . . 7 ⊢ (dom f = a ↔ (1st “ f) = a)
11		df-br 4640	. . . . . . . 8 ⊢ (fImage1st a ↔ ⟨f, a⟩ ∈ Image1st )
12	5, 2	brimage 5793	. . . . . . . 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 3219	. . . . 5 ⊢ (⟨f, a⟩ ∈ (( Funs × V) ∩ Image1st ) ↔ (⟨f, a⟩ ∈ ( Funs × V) ∧ ⟨f, a⟩ ∈ Image1st ))
17		df-fn 4790	. . . . 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 4866	. . 3 ⊢ (( Funs × V) ∩ Image1st ) = {⟨f, a⟩ ∣ f Fn a}
20	1, 19	eqtr4i 2376	. 2 ⊢ Fns = (( Funs × V) ∩ Image1st )
21		funsex 5828	. . . 4 ⊢ Funs ∈ V
22		vvex 4109	. . . 4 ⊢ V ∈ V
23	21, 22	xpex 5115	. . 3 ⊢ ( Funs × V) ∈ V
24		1stex 4739	. . . 4 ⊢ 1st ∈ V
25	24	imageex 5801	. . 3 ⊢ Image1st ∈ V
26	23, 25	inex 4105	. 2 ⊢ (( Funs × V) ∩ Image1st ) ∈ V
27	20, 26	eqeltri 2423	1 ⊢ Fns ∈ V

Syntax hints:   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ∩ cin 3208  ⟨cop 4561  {copab 4622   class class class wbr 4639  1^st c1st 4717   “ cima 4722   × cxp 4770  dom cdm 4772  Fun wfun 4775   Fn wfn 4776  Imagecimage 5753   Funs cfuns 5759   Fns cfns 5761

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-2nd 4797  df-txp 5736  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762

This theorem is referenced by:  enex  6031  ovcelem1  6171  ceex  6174