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Theorem nenpw1pwlem1 6084
 Description: Lemma for nenpw1pw 6086. Set up stratification. (Contributed by SF, 10-Mar-2015.)
Hypothesis
Ref Expression
nenpw1pwlem1.1 S = {x A ¬ x (r ‘{x})}
Assertion
Ref Expression
nenpw1pwlem1 (A VS V)
Distinct variable groups:   x,A   x,r
Allowed substitution hints:   A(r)   S(x,r)   V(x,r)

Proof of Theorem nenpw1pwlem1
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 nenpw1pwlem1.1 . . 3 S = {x A ¬ x (r ‘{x})}
2 dfrab2 3530 . . 3 {x A ¬ x (r ‘{x})} = ({x ¬ x (r ‘{x})} ∩ A)
31, 2eqtri 2373 . 2 S = ({x ¬ x (r ‘{x})} ∩ A)
4 vex 2862 . . . . . . 7 x V
54elcompl 3225 . . . . . 6 (x ∼ ⋃1dom ( FullFun r S ) ↔ ¬ x 1dom ( FullFun r S ))
6 eldm2 4899 . . . . . . . 8 ({x} dom ( FullFun r S ) ↔ y{x}, y ( FullFun r S ))
7 elin 3219 . . . . . . . . . 10 ({x}, y ( FullFun r S ) ↔ ({x}, y FullFun r {x}, y S ))
8 snex 4111 . . . . . . . . . . . . 13 {x} V
98brfullfun 5866 . . . . . . . . . . . 12 ({x} FullFun ry ↔ (r ‘{x}) = y)
10 df-br 4640 . . . . . . . . . . . 12 ({x} FullFun ry{x}, y FullFun r)
11 eqcom 2355 . . . . . . . . . . . 12 ((r ‘{x}) = yy = (r ‘{x}))
129, 10, 113bitr3i 266 . . . . . . . . . . 11 ({x}, y FullFun ry = (r ‘{x}))
13 vex 2862 . . . . . . . . . . . 12 y V
144, 13opelssetsn 4760 . . . . . . . . . . 11 ({x}, y S x y)
1512, 14anbi12i 678 . . . . . . . . . 10 (({x}, y FullFun r {x}, y S ) ↔ (y = (r ‘{x}) x y))
167, 15bitri 240 . . . . . . . . 9 ({x}, y ( FullFun r S ) ↔ (y = (r ‘{x}) x y))
1716exbii 1582 . . . . . . . 8 (y{x}, y ( FullFun r S ) ↔ y(y = (r ‘{x}) x y))
186, 17bitri 240 . . . . . . 7 ({x} dom ( FullFun r S ) ↔ y(y = (r ‘{x}) x y))
194eluni1 4173 . . . . . . 7 (x 1dom ( FullFun r S ) ↔ {x} dom ( FullFun r S ))
20 fvex 5339 . . . . . . . 8 (r ‘{x}) V
2120clel3 2977 . . . . . . 7 (x (r ‘{x}) ↔ y(y = (r ‘{x}) x y))
2218, 19, 213bitr4i 268 . . . . . 6 (x 1dom ( FullFun r S ) ↔ x (r ‘{x}))
235, 22xchbinx 301 . . . . 5 (x ∼ ⋃1dom ( FullFun r S ) ↔ ¬ x (r ‘{x}))
2423abbi2i 2464 . . . 4 ∼ ⋃1dom ( FullFun r S ) = {x ¬ x (r ‘{x})}
25 vex 2862 . . . . . . . . 9 r V
2625fullfunex 5860 . . . . . . . 8 FullFun r V
27 ssetex 4744 . . . . . . . 8 S V
2826, 27inex 4105 . . . . . . 7 ( FullFun r S ) V
2928dmex 5106 . . . . . 6 dom ( FullFun r S ) V
3029uni1ex 4293 . . . . 5 1dom ( FullFun r S ) V
3130complex 4104 . . . 4 ∼ ⋃1dom ( FullFun r S ) V
3224, 31eqeltrri 2424 . . 3 {x ¬ x (r ‘{x})} V
33 inexg 4100 . . 3 (({x ¬ x (r ‘{x})} V A V) → ({x ¬ x (r ‘{x})} ∩ A) V)
3432, 33mpan 651 . 2 (A V → ({x ¬ x (r ‘{x})} ∩ A) V)
353, 34syl5eqel 2437 1 (A VS V)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  {crab 2618  Vcvv 2859   ∼ ccompl 3205   ∩ cin 3208  {csn 3737  ⋃1cuni1 4133  ⟨cop 4561   class class class wbr 4639   S csset 4719  dom cdm 4772   ‘cfv 4781   FullFun cfullfun 5767 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-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-fv 4795  df-2nd 4797  df-fullfun 5768 This theorem is referenced by:  nenpw1pwlem2  6085
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