Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > fvsnun1 | GIF version |
Description: The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 5665. (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 |
---|---|
fvsnun1 | ⊢ (𝐺‘𝐴) = 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvsnun.3 | . . . . 5 ⊢ 𝐺 = ({〈𝐴, 𝐵〉} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) | |
2 | 1 | reseq1i 4862 | . . . 4 ⊢ (𝐺 ↾ {𝐴}) = (({〈𝐴, 𝐵〉} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ {𝐴}) |
3 | resundir 4880 | . . . . 5 ⊢ (({〈𝐴, 𝐵〉} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ {𝐴}) = (({〈𝐴, 𝐵〉} ↾ {𝐴}) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴})) | |
4 | incom 3299 | . . . . . . . . 9 ⊢ ((𝐶 ∖ {𝐴}) ∩ {𝐴}) = ({𝐴} ∩ (𝐶 ∖ {𝐴})) | |
5 | disjdif 3466 | . . . . . . . . 9 ⊢ ({𝐴} ∩ (𝐶 ∖ {𝐴})) = ∅ | |
6 | 4, 5 | eqtri 2178 | . . . . . . . 8 ⊢ ((𝐶 ∖ {𝐴}) ∩ {𝐴}) = ∅ |
7 | resdisj 5014 | . . . . . . . 8 ⊢ (((𝐶 ∖ {𝐴}) ∩ {𝐴}) = ∅ → ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴}) = ∅) | |
8 | 6, 7 | ax-mp 5 | . . . . . . 7 ⊢ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴}) = ∅ |
9 | 8 | uneq2i 3258 | . . . . . 6 ⊢ (({〈𝐴, 𝐵〉} ↾ {𝐴}) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴})) = (({〈𝐴, 𝐵〉} ↾ {𝐴}) ∪ ∅) |
10 | un0 3427 | . . . . . 6 ⊢ (({〈𝐴, 𝐵〉} ↾ {𝐴}) ∪ ∅) = ({〈𝐴, 𝐵〉} ↾ {𝐴}) | |
11 | 9, 10 | eqtri 2178 | . . . . 5 ⊢ (({〈𝐴, 𝐵〉} ↾ {𝐴}) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴})) = ({〈𝐴, 𝐵〉} ↾ {𝐴}) |
12 | 3, 11 | eqtri 2178 | . . . 4 ⊢ (({〈𝐴, 𝐵〉} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ {𝐴}) = ({〈𝐴, 𝐵〉} ↾ {𝐴}) |
13 | 2, 12 | eqtri 2178 | . . 3 ⊢ (𝐺 ↾ {𝐴}) = ({〈𝐴, 𝐵〉} ↾ {𝐴}) |
14 | 13 | fveq1i 5469 | . 2 ⊢ ((𝐺 ↾ {𝐴})‘𝐴) = (({〈𝐴, 𝐵〉} ↾ {𝐴})‘𝐴) |
15 | fvsnun.1 | . . . 4 ⊢ 𝐴 ∈ V | |
16 | 15 | snid 3591 | . . 3 ⊢ 𝐴 ∈ {𝐴} |
17 | fvres 5492 | . . 3 ⊢ (𝐴 ∈ {𝐴} → ((𝐺 ↾ {𝐴})‘𝐴) = (𝐺‘𝐴)) | |
18 | 16, 17 | ax-mp 5 | . 2 ⊢ ((𝐺 ↾ {𝐴})‘𝐴) = (𝐺‘𝐴) |
19 | fvres 5492 | . . . 4 ⊢ (𝐴 ∈ {𝐴} → (({〈𝐴, 𝐵〉} ↾ {𝐴})‘𝐴) = ({〈𝐴, 𝐵〉}‘𝐴)) | |
20 | 16, 19 | ax-mp 5 | . . 3 ⊢ (({〈𝐴, 𝐵〉} ↾ {𝐴})‘𝐴) = ({〈𝐴, 𝐵〉}‘𝐴) |
21 | fvsnun.2 | . . . 4 ⊢ 𝐵 ∈ V | |
22 | 15, 21 | fvsn 5662 | . . 3 ⊢ ({〈𝐴, 𝐵〉}‘𝐴) = 𝐵 |
23 | 20, 22 | eqtri 2178 | . 2 ⊢ (({〈𝐴, 𝐵〉} ↾ {𝐴})‘𝐴) = 𝐵 |
24 | 14, 18, 23 | 3eqtr3i 2186 | 1 ⊢ (𝐺‘𝐴) = 𝐵 |
Colors of variables: wff set class |
Syntax hints: = wceq 1335 ∈ wcel 2128 Vcvv 2712 ∖ cdif 3099 ∪ cun 3100 ∩ cin 3101 ∅c0 3394 {csn 3560 〈cop 3563 ↾ cres 4588 ‘cfv 5170 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-res 4598 df-iota 5135 df-fun 5172 df-fv 5178 |
This theorem is referenced by: fac0 10602 |
Copyright terms: Public domain | W3C validator |