Theorem fvsnun1 5447

Description: The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 5448. (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
fvsnun1	⊢ (G ‘A) = B

Proof of Theorem fvsnun1

Step	Hyp	Ref	Expression
1		fvsnun.1	. . . 4 ⊢ A ∈ V
2	1	snid 3760	. . 3 ⊢ A ∈ {A}
3		fvres 5342	. . 3 ⊢ (A ∈ {A} → ((G ↾ {A}) ‘A) = (G ‘A))
4	2, 3	ax-mp 8	. 2 ⊢ ((G ↾ {A}) ‘A) = (G ‘A)
5		fvsnun.3	. . . . . 6 ⊢ G = ({⟨A, B⟩} ∪ (F ↾ (C ∖ {A})))
6	5	reseq1i 4930	. . . . 5 ⊢ (G ↾ {A}) = (({⟨A, B⟩} ∪ (F ↾ (C ∖ {A}))) ↾ {A})
7		resundir 4982	. . . . 5 ⊢ (({⟨A, B⟩} ∪ (F ↾ (C ∖ {A}))) ↾ {A}) = (({⟨A, B⟩} ↾ {A}) ∪ ((F ↾ (C ∖ {A})) ↾ {A}))
8		incom 3448	. . . . . . . . 9 ⊢ ((C ∖ {A}) ∩ {A}) = ({A} ∩ (C ∖ {A}))
9		disjdif 3622	. . . . . . . . 9 ⊢ ({A} ∩ (C ∖ {A})) = ∅
10	8, 9	eqtri 2373	. . . . . . . 8 ⊢ ((C ∖ {A}) ∩ {A}) = ∅
11		resdisj 5050	. . . . . . . 8 ⊢ (((C ∖ {A}) ∩ {A}) = ∅ → ((F ↾ (C ∖ {A})) ↾ {A}) = ∅)
12	10, 11	ax-mp 8	. . . . . . 7 ⊢ ((F ↾ (C ∖ {A})) ↾ {A}) = ∅
13	12	uneq2i 3415	. . . . . 6 ⊢ (({⟨A, B⟩} ↾ {A}) ∪ ((F ↾ (C ∖ {A})) ↾ {A})) = (({⟨A, B⟩} ↾ {A}) ∪ ∅)
14		un0 3575	. . . . . 6 ⊢ (({⟨A, B⟩} ↾ {A}) ∪ ∅) = ({⟨A, B⟩} ↾ {A})
15	13, 14	eqtri 2373	. . . . 5 ⊢ (({⟨A, B⟩} ↾ {A}) ∪ ((F ↾ (C ∖ {A})) ↾ {A})) = ({⟨A, B⟩} ↾ {A})
16	6, 7, 15	3eqtri 2377	. . . 4 ⊢ (G ↾ {A}) = ({⟨A, B⟩} ↾ {A})
17	16	fveq1i 5329	. . 3 ⊢ ((G ↾ {A}) ‘A) = (({⟨A, B⟩} ↾ {A}) ‘A)
18		fvres 5342	. . . 4 ⊢ (A ∈ {A} → (({⟨A, B⟩} ↾ {A}) ‘A) = ({⟨A, B⟩} ‘A))
19	2, 18	ax-mp 8	. . 3 ⊢ (({⟨A, B⟩} ↾ {A}) ‘A) = ({⟨A, B⟩} ‘A)
20		fvsnun.2	. . . 4 ⊢ B ∈ V
21	1, 20	fvsn 5445	. . 3 ⊢ ({⟨A, B⟩} ‘A) = B
22	17, 19, 21	3eqtri 2377	. 2 ⊢ ((G ↾ {A}) ‘A) = B
23	4, 22	eqtr3i 2375	1 ⊢ (G ‘A) = B

Syntax hints:   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ∖ cdif 3206   ∪ cun 3207   ∩ cin 3208  ∅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-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-fv 4795

This theorem is referenced by: (None)