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Theorem fvunsn 5444
 Description: Remove an ordered pair not participating in a function value. (Contributed by set.mm contributors, 1-Oct-2013.) (Revised by Mario Carneiro, 28-May-2014.)
Assertion
Ref Expression
fvunsn (BD → ((A ∪ {B, C}) ‘D) = (AD))

Proof of Theorem fvunsn
StepHypRef Expression
1 resundir 4982 . . . 4 ((A ∪ {B, C}) {D}) = ((A {D}) ∪ ({B, C} {D}))
2 elsni 3757 . . . . . . . 8 (B {D} → B = D)
32necon3ai 2556 . . . . . . 7 (BD → ¬ B {D})
4 ressnop0 5436 . . . . . . 7 B {D} → ({B, C} {D}) = )
53, 4syl 15 . . . . . 6 (BD → ({B, C} {D}) = )
65uneq2d 3418 . . . . 5 (BD → ((A {D}) ∪ ({B, C} {D})) = ((A {D}) ∪ ))
7 un0 3575 . . . . 5 ((A {D}) ∪ ) = (A {D})
86, 7syl6eq 2401 . . . 4 (BD → ((A {D}) ∪ ({B, C} {D})) = (A {D}))
91, 8syl5eq 2397 . . 3 (BD → ((A ∪ {B, C}) {D}) = (A {D}))
109fveq1d 5330 . 2 (BD → (((A ∪ {B, C}) {D}) ‘D) = ((A {D}) ‘D))
11 snidg 3758 . . . 4 (D V → D {D})
12 fvres 5342 . . . 4 (D {D} → (((A ∪ {B, C}) {D}) ‘D) = ((A ∪ {B, C}) ‘D))
1311, 12syl 15 . . 3 (D V → (((A ∪ {B, C}) {D}) ‘D) = ((A ∪ {B, C}) ‘D))
14 fvprc 5325 . . . 4 D V → (((A ∪ {B, C}) {D}) ‘D) = )
15 fvprc 5325 . . . 4 D V → ((A ∪ {B, C}) ‘D) = )
1614, 15eqtr4d 2388 . . 3 D V → (((A ∪ {B, C}) {D}) ‘D) = ((A ∪ {B, C}) ‘D))
1713, 16pm2.61i 156 . 2 (((A ∪ {B, C}) {D}) ‘D) = ((A ∪ {B, C}) ‘D)
18 fvres 5342 . . . 4 (D {D} → ((A {D}) ‘D) = (AD))
1911, 18syl 15 . . 3 (D V → ((A {D}) ‘D) = (AD))
20 fvprc 5325 . . . 4 D V → ((A {D}) ‘D) = )
21 fvprc 5325 . . . 4 D V → (AD) = )
2220, 21eqtr4d 2388 . . 3 D V → ((A {D}) ‘D) = (AD))
2319, 22pm2.61i 156 . 2 ((A {D}) ‘D) = (AD)
2410, 17, 233eqtr3g 2408 1 (BD → ((A ∪ {B, C}) ‘D) = (AD))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  Vcvv 2859   ∪ cun 3207  ∅c0 3550  {csn 3737  ⟨cop 4561   ↾ cres 4774   ‘cfv 4781 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-xp 4784  df-res 4788  df-fv 4795 This theorem is referenced by:  fvpr1  5449
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