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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.
StepHypRef 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))
42, 3mpbiran2 885 . . . . . . 7 (f, a ( Funs × V) ↔ f Funs )
5 vex 2862 . . . . . . . 8 f V
65elfuns 5829 . . . . . . 7 (f Funs ↔ Fun f)
74, 6bitri 240 . . . . . 6 (f, a ( Funs × V) ↔ Fun f)
8 eqcom 2355 . . . . . . 7 ((1stf) = aa = (1stf))
9 dfdm4 5507 . . . . . . . 8 dom f = (1stf)
109eqeq1i 2360 . . . . . . 7 (dom f = a ↔ (1stf) = a)
11 df-br 4640 . . . . . . . 8 (fImage1st af, a Image1st )
125, 2brimage 5793 . . . . . . . 8 (fImage1st aa = (1stf))
1311, 12bitr3i 242 . . . . . . 7 (f, a Image1sta = (1stf))
148, 10, 133bitr4ri 269 . . . . . 6 (f, a Image1st ↔ dom f = a)
157, 14anbi12i 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))
1815, 16, 173bitr4i 268 . . . 4 (f, a (( Funs × V) ∩ Image1st ) ↔ f Fn a)
1918opabbi2i 4866 . . 3 (( Funs × V) ∩ Image1st ) = {f, a f Fn a}
201, 19eqtr4i 2376 . 2 Fns = (( Funs × V) ∩ Image1st )
21 funsex 5828 . . . 4 Funs V
22 vvex 4109 . . . 4 V V
2321, 22xpex 5115 . . 3 ( Funs × V) V
24 1stex 4739 . . . 4 1st V
2524imageex 5801 . . 3 Image1st V
2623, 25inex 4105 . 2 (( Funs × V) ∩ Image1st ) V
2720, 26eqeltri 2423 1 Fns V
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ∩ cin 3208  ⟨cop 4561  {copab 4622   class class class wbr 4639  1st 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
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