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| Mirrors > Home > MPE Home > Th. List > fvsnun1 | 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 7204. (Contributed by NM, 23-Sep-2007.) Put in deduction form. (Revised by BJ, 25-Feb-2023.) | 
| Ref | Expression | 
|---|---|
| fvsnun.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) | 
| fvsnun.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) | 
| fvsnun.3 | ⊢ 𝐺 = ({〈𝐴, 𝐵〉} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) | 
| Ref | Expression | 
|---|---|
| fvsnun1 | ⊢ (𝜑 → (𝐺‘𝐴) = 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fvsnun.3 | . . . . 5 ⊢ 𝐺 = ({〈𝐴, 𝐵〉} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) | |
| 2 | 1 | reseq1i 5992 | . . . 4 ⊢ (𝐺 ↾ {𝐴}) = (({〈𝐴, 𝐵〉} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ {𝐴}) | 
| 3 | resundir 6011 | . . . . 5 ⊢ (({〈𝐴, 𝐵〉} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ {𝐴}) = (({〈𝐴, 𝐵〉} ↾ {𝐴}) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴})) | |
| 4 | disjdifr 4472 | . . . . . . . 8 ⊢ ((𝐶 ∖ {𝐴}) ∩ {𝐴}) = ∅ | |
| 5 | resdisj 6188 | . . . . . . . 8 ⊢ (((𝐶 ∖ {𝐴}) ∩ {𝐴}) = ∅ → ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴}) = ∅) | |
| 6 | 4, 5 | ax-mp 5 | . . . . . . 7 ⊢ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴}) = ∅ | 
| 7 | 6 | uneq2i 4164 | . . . . . 6 ⊢ (({〈𝐴, 𝐵〉} ↾ {𝐴}) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴})) = (({〈𝐴, 𝐵〉} ↾ {𝐴}) ∪ ∅) | 
| 8 | un0 4393 | . . . . . 6 ⊢ (({〈𝐴, 𝐵〉} ↾ {𝐴}) ∪ ∅) = ({〈𝐴, 𝐵〉} ↾ {𝐴}) | |
| 9 | 7, 8 | eqtri 2764 | . . . . 5 ⊢ (({〈𝐴, 𝐵〉} ↾ {𝐴}) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴})) = ({〈𝐴, 𝐵〉} ↾ {𝐴}) | 
| 10 | 3, 9 | eqtri 2764 | . . . 4 ⊢ (({〈𝐴, 𝐵〉} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ {𝐴}) = ({〈𝐴, 𝐵〉} ↾ {𝐴}) | 
| 11 | 2, 10 | eqtri 2764 | . . 3 ⊢ (𝐺 ↾ {𝐴}) = ({〈𝐴, 𝐵〉} ↾ {𝐴}) | 
| 12 | 11 | fveq1i 6906 | . 2 ⊢ ((𝐺 ↾ {𝐴})‘𝐴) = (({〈𝐴, 𝐵〉} ↾ {𝐴})‘𝐴) | 
| 13 | fvsnun.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
| 14 | snidg 4659 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴}) | |
| 15 | 13, 14 | syl 17 | . . 3 ⊢ (𝜑 → 𝐴 ∈ {𝐴}) | 
| 16 | 15 | fvresd 6925 | . 2 ⊢ (𝜑 → ((𝐺 ↾ {𝐴})‘𝐴) = (𝐺‘𝐴)) | 
| 17 | 15 | fvresd 6925 | . . 3 ⊢ (𝜑 → (({〈𝐴, 𝐵〉} ↾ {𝐴})‘𝐴) = ({〈𝐴, 𝐵〉}‘𝐴)) | 
| 18 | fvsnun.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
| 19 | fvsng 7201 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({〈𝐴, 𝐵〉}‘𝐴) = 𝐵) | |
| 20 | 13, 18, 19 | syl2anc 584 | . . 3 ⊢ (𝜑 → ({〈𝐴, 𝐵〉}‘𝐴) = 𝐵) | 
| 21 | 17, 20 | eqtrd 2776 | . 2 ⊢ (𝜑 → (({〈𝐴, 𝐵〉} ↾ {𝐴})‘𝐴) = 𝐵) | 
| 22 | 12, 16, 21 | 3eqtr3a 2800 | 1 ⊢ (𝜑 → (𝐺‘𝐴) = 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ∖ cdif 3947 ∪ cun 3948 ∩ cin 3949 ∅c0 4332 {csn 4625 〈cop 4631 ↾ cres 5686 ‘cfv 6560 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-res 5696 df-iota 6513 df-fun 6562 df-fv 6568 | 
| This theorem is referenced by: fac0 14316 ruclem4 16271 satfv1lem 35368 | 
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