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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-fvmptunsn2 | Structured version Visualization version GIF version |
Description: Value of a function expressed as a union of a mapsto expression and a singleton on a couple (with disjoint domain) at a point in the domain of the mapsto construction. (Contributed by BJ, 18-Mar-2023.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-fvmptunsn.un | ⊢ (𝜑 → 𝐹 = ((𝑥 ∈ 𝐴 ↦ 𝐵) ∪ {⟨𝐶, 𝐷⟩})) |
bj-fvmptunsn.nel | ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴) |
bj-fvmptunsn2.el | ⊢ (𝜑 → 𝐸 ∈ 𝐴) |
bj-fvmptunsn2.ex | ⊢ (𝜑 → 𝐺 ∈ 𝑉) |
bj-fvmptunsn2.is | ⊢ ((𝜑 ∧ 𝑥 = 𝐸) → 𝐵 = 𝐺) |
Ref | Expression |
---|---|
bj-fvmptunsn2 | ⊢ (𝜑 → (𝐹‘𝐸) = 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-fvmptunsn.un | . . 3 ⊢ (𝜑 → 𝐹 = ((𝑥 ∈ 𝐴 ↦ 𝐵) ∪ {⟨𝐶, 𝐷⟩})) | |
2 | bj-fvmptunsn2.el | . . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝐴) | |
3 | bj-fvmptunsn.nel | . . . 4 ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴) | |
4 | nelneq 2849 | . . . 4 ⊢ ((𝐸 ∈ 𝐴 ∧ ¬ 𝐶 ∈ 𝐴) → ¬ 𝐸 = 𝐶) | |
5 | 2, 3, 4 | syl2anc 582 | . . 3 ⊢ (𝜑 → ¬ 𝐸 = 𝐶) |
6 | 1, 5 | bj-fununsn1 36785 | . 2 ⊢ (𝜑 → (𝐹‘𝐸) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝐸)) |
7 | eqidd 2726 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)) | |
8 | bj-fvmptunsn2.is | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐸) → 𝐵 = 𝐺) | |
9 | bj-fvmptunsn2.ex | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑉) | |
10 | 7, 8, 2, 9 | fvmptd 7005 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝐸) = 𝐺) |
11 | 6, 10 | eqtrd 2765 | 1 ⊢ (𝜑 → (𝐹‘𝐸) = 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∪ cun 3939 {csn 4625 ⟨cop 4631 ↦ cmpt 5227 ‘cfv 6543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5295 ax-nul 5302 ax-pr 5424 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4320 df-if 4526 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6495 df-fun 6545 df-fv 6551 |
This theorem is referenced by: (None) |
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