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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 5382. (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 4626 | . . . 4 ⊢ (𝐺 ↾ {𝐴}) = (({〈𝐴, 𝐵〉} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ {𝐴}) |
3 | resundir 4644 | . . . . 5 ⊢ (({〈𝐴, 𝐵〉} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ {𝐴}) = (({〈𝐴, 𝐵〉} ↾ {𝐴}) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴})) | |
4 | incom 3158 | . . . . . . . . 9 ⊢ ((𝐶 ∖ {𝐴}) ∩ {𝐴}) = ({𝐴} ∩ (𝐶 ∖ {𝐴})) | |
5 | disjdif 3316 | . . . . . . . . 9 ⊢ ({𝐴} ∩ (𝐶 ∖ {𝐴})) = ∅ | |
6 | 4, 5 | eqtri 2101 | . . . . . . . 8 ⊢ ((𝐶 ∖ {𝐴}) ∩ {𝐴}) = ∅ |
7 | resdisj 4771 | . . . . . . . 8 ⊢ (((𝐶 ∖ {𝐴}) ∩ {𝐴}) = ∅ → ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴}) = ∅) | |
8 | 6, 7 | ax-mp 7 | . . . . . . 7 ⊢ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴}) = ∅ |
9 | 8 | uneq2i 3123 | . . . . . 6 ⊢ (({〈𝐴, 𝐵〉} ↾ {𝐴}) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴})) = (({〈𝐴, 𝐵〉} ↾ {𝐴}) ∪ ∅) |
10 | un0 3278 | . . . . . 6 ⊢ (({〈𝐴, 𝐵〉} ↾ {𝐴}) ∪ ∅) = ({〈𝐴, 𝐵〉} ↾ {𝐴}) | |
11 | 9, 10 | eqtri 2101 | . . . . 5 ⊢ (({〈𝐴, 𝐵〉} ↾ {𝐴}) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴})) = ({〈𝐴, 𝐵〉} ↾ {𝐴}) |
12 | 3, 11 | eqtri 2101 | . . . 4 ⊢ (({〈𝐴, 𝐵〉} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ {𝐴}) = ({〈𝐴, 𝐵〉} ↾ {𝐴}) |
13 | 2, 12 | eqtri 2101 | . . 3 ⊢ (𝐺 ↾ {𝐴}) = ({〈𝐴, 𝐵〉} ↾ {𝐴}) |
14 | 13 | fveq1i 5199 | . 2 ⊢ ((𝐺 ↾ {𝐴})‘𝐴) = (({〈𝐴, 𝐵〉} ↾ {𝐴})‘𝐴) |
15 | fvsnun.1 | . . . 4 ⊢ 𝐴 ∈ V | |
16 | 15 | snid 3425 | . . 3 ⊢ 𝐴 ∈ {𝐴} |
17 | fvres 5219 | . . 3 ⊢ (𝐴 ∈ {𝐴} → ((𝐺 ↾ {𝐴})‘𝐴) = (𝐺‘𝐴)) | |
18 | 16, 17 | ax-mp 7 | . 2 ⊢ ((𝐺 ↾ {𝐴})‘𝐴) = (𝐺‘𝐴) |
19 | fvres 5219 | . . . 4 ⊢ (𝐴 ∈ {𝐴} → (({〈𝐴, 𝐵〉} ↾ {𝐴})‘𝐴) = ({〈𝐴, 𝐵〉}‘𝐴)) | |
20 | 16, 19 | ax-mp 7 | . . 3 ⊢ (({〈𝐴, 𝐵〉} ↾ {𝐴})‘𝐴) = ({〈𝐴, 𝐵〉}‘𝐴) |
21 | fvsnun.2 | . . . 4 ⊢ 𝐵 ∈ V | |
22 | 15, 21 | fvsn 5379 | . . 3 ⊢ ({〈𝐴, 𝐵〉}‘𝐴) = 𝐵 |
23 | 20, 22 | eqtri 2101 | . 2 ⊢ (({〈𝐴, 𝐵〉} ↾ {𝐴})‘𝐴) = 𝐵 |
24 | 14, 18, 23 | 3eqtr3i 2109 | 1 ⊢ (𝐺‘𝐴) = 𝐵 |
Colors of variables: wff set class |
Syntax hints: = wceq 1284 ∈ wcel 1433 Vcvv 2601 ∖ cdif 2970 ∪ cun 2971 ∩ cin 2972 ∅c0 3251 {csn 3398 〈cop 3401 ↾ cres 4365 ‘cfv 4922 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-res 4375 df-iota 4887 df-fun 4924 df-fv 4930 |
This theorem is referenced by: fac0 9655 |
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