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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 7130. (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 5934 | . . . 4 ⊢ (𝐺 ↾ {𝐴}) = (({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ {𝐴}) |
3 | resundir 5953 | . . . . 5 ⊢ (({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ {𝐴}) = (({⟨𝐴, 𝐵⟩} ↾ {𝐴}) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴})) | |
4 | disjdifr 4433 | . . . . . . . 8 ⊢ ((𝐶 ∖ {𝐴}) ∩ {𝐴}) = ∅ | |
5 | resdisj 6122 | . . . . . . . 8 ⊢ (((𝐶 ∖ {𝐴}) ∩ {𝐴}) = ∅ → ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴}) = ∅) | |
6 | 4, 5 | ax-mp 5 | . . . . . . 7 ⊢ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴}) = ∅ |
7 | 6 | uneq2i 4121 | . . . . . 6 ⊢ (({⟨𝐴, 𝐵⟩} ↾ {𝐴}) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴})) = (({⟨𝐴, 𝐵⟩} ↾ {𝐴}) ∪ ∅) |
8 | un0 4351 | . . . . . 6 ⊢ (({⟨𝐴, 𝐵⟩} ↾ {𝐴}) ∪ ∅) = ({⟨𝐴, 𝐵⟩} ↾ {𝐴}) | |
9 | 7, 8 | eqtri 2761 | . . . . 5 ⊢ (({⟨𝐴, 𝐵⟩} ↾ {𝐴}) ∪ ((𝐹 ↾ (𝐶 ∖ {𝐴})) ↾ {𝐴})) = ({⟨𝐴, 𝐵⟩} ↾ {𝐴}) |
10 | 3, 9 | eqtri 2761 | . . . 4 ⊢ (({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴}))) ↾ {𝐴}) = ({⟨𝐴, 𝐵⟩} ↾ {𝐴}) |
11 | 2, 10 | eqtri 2761 | . . 3 ⊢ (𝐺 ↾ {𝐴}) = ({⟨𝐴, 𝐵⟩} ↾ {𝐴}) |
12 | 11 | fveq1i 6844 | . 2 ⊢ ((𝐺 ↾ {𝐴})‘𝐴) = (({⟨𝐴, 𝐵⟩} ↾ {𝐴})‘𝐴) |
13 | fvsnun.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
14 | snidg 4621 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴}) | |
15 | 13, 14 | syl 17 | . . 3 ⊢ (𝜑 → 𝐴 ∈ {𝐴}) |
16 | 15 | fvresd 6863 | . 2 ⊢ (𝜑 → ((𝐺 ↾ {𝐴})‘𝐴) = (𝐺‘𝐴)) |
17 | 15 | fvresd 6863 | . . 3 ⊢ (𝜑 → (({⟨𝐴, 𝐵⟩} ↾ {𝐴})‘𝐴) = ({⟨𝐴, 𝐵⟩}‘𝐴)) |
18 | fvsnun.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
19 | fvsng 7127 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵) | |
20 | 13, 18, 19 | syl2anc 585 | . . 3 ⊢ (𝜑 → ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵) |
21 | 17, 20 | eqtrd 2773 | . 2 ⊢ (𝜑 → (({⟨𝐴, 𝐵⟩} ↾ {𝐴})‘𝐴) = 𝐵) |
22 | 12, 16, 21 | 3eqtr3a 2797 | 1 ⊢ (𝜑 → (𝐺‘𝐴) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ∖ cdif 3908 ∪ cun 3909 ∩ cin 3910 ∅c0 4283 {csn 4587 ⟨cop 4593 ↾ cres 5636 ‘cfv 6497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-res 5646 df-iota 6449 df-fun 6499 df-fv 6505 |
This theorem is referenced by: fac0 14182 ruclem4 16121 satfv1lem 34013 |
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