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Mirrors > Home > MPE Home > Th. List > fvsnun2 | Structured version Visualization version 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 6353. (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 5313 | . . . 4 ⊢ (𝐺 ↾ (𝐶 ∖ {𝐴})) = (({〈𝐴, 𝐵〉} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ (𝐶 ∖ {𝐴})) |
3 | resundir 5331 | . . . 4 ⊢ (({〈𝐴, 𝐵〉} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ (𝐶 ∖ {𝐴})) = (({〈𝐴, 𝐵〉} ↾ (𝐶 ∖ {𝐴})) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴}))) | |
4 | disjdif 3992 | . . . . . . 7 ⊢ ({𝐴} ∩ (𝐶 ∖ {𝐴})) = ∅ | |
5 | fvsnun.1 | . . . . . . . . 9 ⊢ 𝐴 ∈ V | |
6 | fvsnun.2 | . . . . . . . . 9 ⊢ 𝐵 ∈ V | |
7 | 5, 6 | fnsn 5860 | . . . . . . . 8 ⊢ {〈𝐴, 𝐵〉} Fn {𝐴} |
8 | fnresdisj 5915 | . . . . . . . 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 219 | . . . . . 6 ⊢ ({〈𝐴, 𝐵〉} ↾ (𝐶 ∖ {𝐴})) = ∅ |
11 | residm 5350 | . . . . . 6 ⊢ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴})) = (𝐹 ↾ (𝐶 ∖ {𝐴})) | |
12 | 10, 11 | uneq12i 3727 | . . . . 5 ⊢ (({〈𝐴, 𝐵〉} ↾ (𝐶 ∖ {𝐴})) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴}))) = (∅ ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) |
13 | uncom 3719 | . . . . 5 ⊢ (∅ ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) = ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ ∅) | |
14 | un0 3919 | . . . . 5 ⊢ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ ∅) = (𝐹 ↾ (𝐶 ∖ {𝐴})) | |
15 | 12, 13, 14 | 3eqtri 2636 | . . . 4 ⊢ (({〈𝐴, 𝐵〉} ↾ (𝐶 ∖ {𝐴})) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ (𝐶 ∖ {𝐴}))) = (𝐹 ↾ (𝐶 ∖ {𝐴})) |
16 | 2, 3, 15 | 3eqtri 2636 | . . 3 ⊢ (𝐺 ↾ (𝐶 ∖ {𝐴})) = (𝐹 ↾ (𝐶 ∖ {𝐴})) |
17 | 16 | fveq1i 6104 | . 2 ⊢ ((𝐺 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷) |
18 | fvres 6117 | . 2 ⊢ (𝐷 ∈ (𝐶 ∖ {𝐴}) → ((𝐺 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = (𝐺‘𝐷)) | |
19 | fvres 6117 | . 2 ⊢ (𝐷 ∈ (𝐶 ∖ {𝐴}) → ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = (𝐹‘𝐷)) | |
20 | 17, 18, 19 | 3eqtr3a 2668 | 1 ⊢ (𝐷 ∈ (𝐶 ∖ {𝐴}) → (𝐺‘𝐷) = (𝐹‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∖ cdif 3537 ∪ cun 3538 ∩ cin 3539 ∅c0 3874 {csn 4125 〈cop 4131 ↾ cres 5040 Fn wfn 5799 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-res 5050 df-iota 5768 df-fun 5806 df-fn 5807 df-fv 5812 |
This theorem is referenced by: facnn 12924 |
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