![]() |
Mathbox for BJ |
< Previous
Next >
Nearby theorems |
|
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 2852 | . . . 4 ⊢ ((𝐸 ∈ 𝐴 ∧ ¬ 𝐶 ∈ 𝐴) → ¬ 𝐸 = 𝐶) | |
5 | 2, 3, 4 | syl2anc 583 | . . 3 ⊢ (𝜑 → ¬ 𝐸 = 𝐶) |
6 | 1, 5 | bj-fununsn1 36655 | . 2 ⊢ (𝜑 → (𝐹‘𝐸) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝐸)) |
7 | eqidd 2728 | . . 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 7006 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝐸) = 𝐺) |
11 | 6, 10 | eqtrd 2767 | 1 ⊢ (𝜑 → (𝐹‘𝐸) = 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∪ cun 3942 {csn 4624 ⟨cop 4630 ↦ cmpt 5225 ‘cfv 6542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fv 6550 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |