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Theorem fvsnun2 5448
 Description: The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 5447. (Contributed by set.mm contributors, 23-Sep-2007.)
Hypotheses
Ref Expression
fvsnun.1 A V
fvsnun.2 B V
fvsnun.3 G = ({A, B} ∪ (F (C {A})))
Assertion
Ref Expression
fvsnun2 (D (C {A}) → (GD) = (FD))

Proof of Theorem fvsnun2
StepHypRef Expression
1 fvres 5342 . 2 (D (C {A}) → ((G (C {A})) ‘D) = (GD))
2 fvsnun.3 . . . . . 6 G = ({A, B} ∪ (F (C {A})))
32reseq1i 4930 . . . . 5 (G (C {A})) = (({A, B} ∪ (F (C {A}))) (C {A}))
4 resundir 4982 . . . . 5 (({A, B} ∪ (F (C {A}))) (C {A})) = (({A, B} (C {A})) ∪ ((F (C {A})) (C {A})))
5 disjdif 3622 . . . . . . . 8 ({A} ∩ (C {A})) =
6 fvsnun.1 . . . . . . . . . 10 A V
7 fvsnun.2 . . . . . . . . . 10 B V
86, 7fnsn 5152 . . . . . . . . 9 {A, B} Fn {A}
9 fnresdisj 5193 . . . . . . . . 9 ({A, B} Fn {A} → (({A} ∩ (C {A})) = ↔ ({A, B} (C {A})) = ))
108, 9ax-mp 8 . . . . . . . 8 (({A} ∩ (C {A})) = ↔ ({A, B} (C {A})) = )
115, 10mpbi 199 . . . . . . 7 ({A, B} (C {A})) =
12 residm 4994 . . . . . . 7 ((F (C {A})) (C {A})) = (F (C {A}))
1311, 12uneq12i 3416 . . . . . 6 (({A, B} (C {A})) ∪ ((F (C {A})) (C {A}))) = ( ∪ (F (C {A})))
14 uncom 3408 . . . . . 6 ( ∪ (F (C {A}))) = ((F (C {A})) ∪ )
15 un0 3575 . . . . . 6 ((F (C {A})) ∪ ) = (F (C {A}))
1613, 14, 153eqtri 2377 . . . . 5 (({A, B} (C {A})) ∪ ((F (C {A})) (C {A}))) = (F (C {A}))
173, 4, 163eqtri 2377 . . . 4 (G (C {A})) = (F (C {A}))
1817fveq1i 5329 . . 3 ((G (C {A})) ‘D) = ((F (C {A})) ‘D)
19 fvres 5342 . . 3 (D (C {A}) → ((F (C {A})) ‘D) = (FD))
2018, 19syl5eq 2397 . 2 (D (C {A}) → ((G (C {A})) ‘D) = (FD))
211, 20eqtr3d 2387 1 (D (C {A}) → (GD) = (FD))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ∖ cdif 3206   ∪ cun 3207   ∩ cin 3208  ∅c0 3550  {csn 3737  ⟨cop 4561   ↾ cres 4774   Fn wfn 4776   ‘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-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-fv 4795 This theorem is referenced by: (None)
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