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Mirrors > Home > ILE Home > Th. List > fvsnun2 | GIF version |
Description: The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 5625. (Contributed by NM, 23-Sep-2007.) |
Ref | Expression |
---|---|
fvsnun.1 | ⊢ 𝐴 ∈ V |
fvsnun.2 | ⊢ 𝐵 ∈ V |
fvsnun.3 | ⊢ 𝐺 = ({〈𝐴, 𝐵〉} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) |
Ref | Expression |
---|---|
fvsnun2 | ⊢ (𝐷 ∈ (𝐶 ∖ {𝐴}) → (𝐺‘𝐷) = (𝐹‘𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvsnun.3 | . . . . 5 ⊢ 𝐺 = ({〈𝐴, 𝐵〉} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) | |
2 | 1 | reseq1i 4823 | . . . 4 ⊢ (𝐺 ↾ (𝐶 ∖ {𝐴})) = (({〈𝐴, 𝐵〉} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ (𝐶 ∖ {𝐴})) |
3 | resundir 4841 | . . . 4 ⊢ (({〈𝐴, 𝐵〉} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ (𝐶 ∖ {𝐴})) = (({〈𝐴, 𝐵〉} ↾ (𝐶 ∖ {𝐴})) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴}))) | |
4 | disjdif 3440 | . . . . . . 7 ⊢ ({𝐴} ∩ (𝐶 ∖ {𝐴})) = ∅ | |
5 | fvsnun.1 | . . . . . . . . 9 ⊢ 𝐴 ∈ V | |
6 | fvsnun.2 | . . . . . . . . 9 ⊢ 𝐵 ∈ V | |
7 | 5, 6 | fnsn 5185 | . . . . . . . 8 ⊢ {〈𝐴, 𝐵〉} Fn {𝐴} |
8 | fnresdisj 5241 | . . . . . . . 8 ⊢ ({〈𝐴, 𝐵〉} Fn {𝐴} → (({𝐴} ∩ (𝐶 ∖ {𝐴})) = ∅ ↔ ({〈𝐴, 𝐵〉} ↾ (𝐶 ∖ {𝐴})) = ∅)) | |
9 | 7, 8 | ax-mp 5 | . . . . . . 7 ⊢ (({𝐴} ∩ (𝐶 ∖ {𝐴})) = ∅ ↔ ({〈𝐴, 𝐵〉} ↾ (𝐶 ∖ {𝐴})) = ∅) |
10 | 4, 9 | mpbi 144 | . . . . . 6 ⊢ ({〈𝐴, 𝐵〉} ↾ (𝐶 ∖ {𝐴})) = ∅ |
11 | residm 4859 | . . . . . 6 ⊢ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴})) = (𝐹 ↾ (𝐶 ∖ {𝐴})) | |
12 | 10, 11 | uneq12i 3233 | . . . . 5 ⊢ (({〈𝐴, 𝐵〉} ↾ (𝐶 ∖ {𝐴})) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴}))) = (∅ ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) |
13 | uncom 3225 | . . . . 5 ⊢ (∅ ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) = ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ ∅) | |
14 | un0 3401 | . . . . 5 ⊢ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ ∅) = (𝐹 ↾ (𝐶 ∖ {𝐴})) | |
15 | 12, 13, 14 | 3eqtri 2165 | . . . 4 ⊢ (({〈𝐴, 𝐵〉} ↾ (𝐶 ∖ {𝐴})) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴}))) = (𝐹 ↾ (𝐶 ∖ {𝐴})) |
16 | 2, 3, 15 | 3eqtri 2165 | . . 3 ⊢ (𝐺 ↾ (𝐶 ∖ {𝐴})) = (𝐹 ↾ (𝐶 ∖ {𝐴})) |
17 | 16 | fveq1i 5430 | . 2 ⊢ ((𝐺 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷) |
18 | fvres 5453 | . 2 ⊢ (𝐷 ∈ (𝐶 ∖ {𝐴}) → ((𝐺 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = (𝐺‘𝐷)) | |
19 | fvres 5453 | . 2 ⊢ (𝐷 ∈ (𝐶 ∖ {𝐴}) → ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = (𝐹‘𝐷)) | |
20 | 17, 18, 19 | 3eqtr3a 2197 | 1 ⊢ (𝐷 ∈ (𝐶 ∖ {𝐴}) → (𝐺‘𝐷) = (𝐹‘𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1332 ∈ wcel 1481 Vcvv 2689 ∖ cdif 3073 ∪ cun 3074 ∩ cin 3075 ∅c0 3368 {csn 3532 〈cop 3535 ↾ cres 4549 Fn wfn 5126 ‘cfv 5131 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-res 4559 df-iota 5096 df-fun 5133 df-fn 5134 df-fv 5139 |
This theorem is referenced by: facnn 10505 |
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