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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 5360. (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 4608 | . . . 4 ⊢ (𝐺 ↾ (𝐶 ∖ {𝐴})) = (({〈𝐴, 𝐵〉} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ (𝐶 ∖ {𝐴})) |
3 | resundir 4626 | . . . 4 ⊢ (({〈𝐴, 𝐵〉} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ (𝐶 ∖ {𝐴})) = (({〈𝐴, 𝐵〉} ↾ (𝐶 ∖ {𝐴})) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴}))) | |
4 | disjdif 3296 | . . . . . . 7 ⊢ ({𝐴} ∩ (𝐶 ∖ {𝐴})) = ∅ | |
5 | fvsnun.1 | . . . . . . . . 9 ⊢ 𝐴 ∈ V | |
6 | fvsnun.2 | . . . . . . . . 9 ⊢ 𝐵 ∈ V | |
7 | 5, 6 | fnsn 4953 | . . . . . . . 8 ⊢ {〈𝐴, 𝐵〉} Fn {𝐴} |
8 | fnresdisj 5009 | . . . . . . . 8 ⊢ ({〈𝐴, 𝐵〉} Fn {𝐴} → (({𝐴} ∩ (𝐶 ∖ {𝐴})) = ∅ ↔ ({〈𝐴, 𝐵〉} ↾ (𝐶 ∖ {𝐴})) = ∅)) | |
9 | 7, 8 | ax-mp 7 | . . . . . . 7 ⊢ (({𝐴} ∩ (𝐶 ∖ {𝐴})) = ∅ ↔ ({〈𝐴, 𝐵〉} ↾ (𝐶 ∖ {𝐴})) = ∅) |
10 | 4, 9 | mpbi 133 | . . . . . 6 ⊢ ({〈𝐴, 𝐵〉} ↾ (𝐶 ∖ {𝐴})) = ∅ |
11 | residm 4642 | . . . . . 6 ⊢ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴})) = (𝐹 ↾ (𝐶 ∖ {𝐴})) | |
12 | 10, 11 | uneq12i 3095 | . . . . 5 ⊢ (({〈𝐴, 𝐵〉} ↾ (𝐶 ∖ {𝐴})) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴}))) = (∅ ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) |
13 | uncom 3087 | . . . . 5 ⊢ (∅ ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) = ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ ∅) | |
14 | un0 3251 | . . . . 5 ⊢ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ ∅) = (𝐹 ↾ (𝐶 ∖ {𝐴})) | |
15 | 12, 13, 14 | 3eqtri 2064 | . . . 4 ⊢ (({〈𝐴, 𝐵〉} ↾ (𝐶 ∖ {𝐴})) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴}))) = (𝐹 ↾ (𝐶 ∖ {𝐴})) |
16 | 2, 3, 15 | 3eqtri 2064 | . . 3 ⊢ (𝐺 ↾ (𝐶 ∖ {𝐴})) = (𝐹 ↾ (𝐶 ∖ {𝐴})) |
17 | 16 | fveq1i 5179 | . 2 ⊢ ((𝐺 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷) |
18 | fvres 5198 | . 2 ⊢ (𝐷 ∈ (𝐶 ∖ {𝐴}) → ((𝐺 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = (𝐺‘𝐷)) | |
19 | fvres 5198 | . 2 ⊢ (𝐷 ∈ (𝐶 ∖ {𝐴}) → ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = (𝐹‘𝐷)) | |
20 | 17, 18, 19 | 3eqtr3a 2096 | 1 ⊢ (𝐷 ∈ (𝐶 ∖ {𝐴}) → (𝐺‘𝐷) = (𝐹‘𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 = wceq 1243 ∈ wcel 1393 Vcvv 2557 ∖ cdif 2914 ∪ cun 2915 ∩ cin 2916 ∅c0 3224 {csn 3375 〈cop 3378 ↾ cres 4347 Fn wfn 4897 ‘cfv 4902 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-res 4357 df-iota 4867 df-fun 4904 df-fn 4905 df-fv 4910 |
This theorem is referenced by: (None) |
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