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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 5586. (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 4785 | . . . 4 ⊢ (𝐺 ↾ {𝐴}) = (({〈𝐴, 𝐵〉} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ {𝐴}) |
3 | resundir 4803 | . . . . 5 ⊢ (({〈𝐴, 𝐵〉} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ {𝐴}) = (({〈𝐴, 𝐵〉} ↾ {𝐴}) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴})) | |
4 | incom 3238 | . . . . . . . . 9 ⊢ ((𝐶 ∖ {𝐴}) ∩ {𝐴}) = ({𝐴} ∩ (𝐶 ∖ {𝐴})) | |
5 | disjdif 3405 | . . . . . . . . 9 ⊢ ({𝐴} ∩ (𝐶 ∖ {𝐴})) = ∅ | |
6 | 4, 5 | eqtri 2138 | . . . . . . . 8 ⊢ ((𝐶 ∖ {𝐴}) ∩ {𝐴}) = ∅ |
7 | resdisj 4937 | . . . . . . . 8 ⊢ (((𝐶 ∖ {𝐴}) ∩ {𝐴}) = ∅ → ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴}) = ∅) | |
8 | 6, 7 | ax-mp 5 | . . . . . . 7 ⊢ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴}) = ∅ |
9 | 8 | uneq2i 3197 | . . . . . 6 ⊢ (({〈𝐴, 𝐵〉} ↾ {𝐴}) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴})) = (({〈𝐴, 𝐵〉} ↾ {𝐴}) ∪ ∅) |
10 | un0 3366 | . . . . . 6 ⊢ (({〈𝐴, 𝐵〉} ↾ {𝐴}) ∪ ∅) = ({〈𝐴, 𝐵〉} ↾ {𝐴}) | |
11 | 9, 10 | eqtri 2138 | . . . . 5 ⊢ (({〈𝐴, 𝐵〉} ↾ {𝐴}) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴})) = ({〈𝐴, 𝐵〉} ↾ {𝐴}) |
12 | 3, 11 | eqtri 2138 | . . . 4 ⊢ (({〈𝐴, 𝐵〉} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ {𝐴}) = ({〈𝐴, 𝐵〉} ↾ {𝐴}) |
13 | 2, 12 | eqtri 2138 | . . 3 ⊢ (𝐺 ↾ {𝐴}) = ({〈𝐴, 𝐵〉} ↾ {𝐴}) |
14 | 13 | fveq1i 5390 | . 2 ⊢ ((𝐺 ↾ {𝐴})‘𝐴) = (({〈𝐴, 𝐵〉} ↾ {𝐴})‘𝐴) |
15 | fvsnun.1 | . . . 4 ⊢ 𝐴 ∈ V | |
16 | 15 | snid 3526 | . . 3 ⊢ 𝐴 ∈ {𝐴} |
17 | fvres 5413 | . . 3 ⊢ (𝐴 ∈ {𝐴} → ((𝐺 ↾ {𝐴})‘𝐴) = (𝐺‘𝐴)) | |
18 | 16, 17 | ax-mp 5 | . 2 ⊢ ((𝐺 ↾ {𝐴})‘𝐴) = (𝐺‘𝐴) |
19 | fvres 5413 | . . . 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 5583 | . . 3 ⊢ ({〈𝐴, 𝐵〉}‘𝐴) = 𝐵 |
23 | 20, 22 | eqtri 2138 | . 2 ⊢ (({〈𝐴, 𝐵〉} ↾ {𝐴})‘𝐴) = 𝐵 |
24 | 14, 18, 23 | 3eqtr3i 2146 | 1 ⊢ (𝐺‘𝐴) = 𝐵 |
Colors of variables: wff set class |
Syntax hints: = wceq 1316 ∈ wcel 1465 Vcvv 2660 ∖ cdif 3038 ∪ cun 3039 ∩ cin 3040 ∅c0 3333 {csn 3497 〈cop 3500 ↾ cres 4511 ‘cfv 5093 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-res 4521 df-iota 5058 df-fun 5095 df-fv 5101 |
This theorem is referenced by: fac0 10442 |
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