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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-fvsnun2 | Structured version Visualization version GIF version |
Description: The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 6922. (Contributed by NM, 23-Sep-2007.) Put in deduction form. (Revised by BJ, 18-Mar-2023.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-fvsnun.un | ⊢ (𝜑 → 𝐺 = ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉})) |
bj-fvsnun2.ex1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
bj-fvsnun2.ex2 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
Ref | Expression |
---|---|
bj-fvsnun2 | ⊢ (𝜑 → (𝐺‘𝐴) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-fvsnun.un | . 2 ⊢ (𝜑 → 𝐺 = ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ {〈𝐴, 𝐵〉})) | |
2 | dmres 5840 | . . . . 5 ⊢ dom (𝐹 ↾ (𝐶 ∖ {𝐴})) = ((𝐶 ∖ {𝐴}) ∩ dom 𝐹) | |
3 | inss1 4155 | . . . . 5 ⊢ ((𝐶 ∖ {𝐴}) ∩ dom 𝐹) ⊆ (𝐶 ∖ {𝐴}) | |
4 | 2, 3 | eqsstri 3949 | . . . 4 ⊢ dom (𝐹 ↾ (𝐶 ∖ {𝐴})) ⊆ (𝐶 ∖ {𝐴}) |
5 | 4 | a1i 11 | . . 3 ⊢ (𝜑 → dom (𝐹 ↾ (𝐶 ∖ {𝐴})) ⊆ (𝐶 ∖ {𝐴})) |
6 | neldifsnd 4686 | . . 3 ⊢ (𝜑 → ¬ 𝐴 ∈ (𝐶 ∖ {𝐴})) | |
7 | 5, 6 | ssneldd 3918 | . 2 ⊢ (𝜑 → ¬ 𝐴 ∈ dom (𝐹 ↾ (𝐶 ∖ {𝐴}))) |
8 | bj-fvsnun2.ex1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
9 | bj-fvsnun2.ex2 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
10 | 1, 7, 8, 9 | bj-fununsn2 34669 | 1 ⊢ (𝜑 → (𝐺‘𝐴) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ∖ cdif 3878 ∪ cun 3879 ∩ cin 3880 ⊆ wss 3881 {csn 4525 〈cop 4531 dom cdm 5519 ↾ cres 5521 ‘cfv 6324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fv 6332 |
This theorem is referenced by: (None) |
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