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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-fvmptunsn1 | 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 the first component of that couple. (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-fvmptunsn1.ex1 | ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
bj-fvmptunsn1.ex2 | ⊢ (𝜑 → 𝐷 ∈ 𝑊) |
Ref | Expression |
---|---|
bj-fvmptunsn1 | ⊢ (𝜑 → (𝐹‘𝐶) = 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-fvmptunsn.un | . 2 ⊢ (𝜑 → 𝐹 = ((𝑥 ∈ 𝐴 ↦ 𝐵) ∪ {⟨𝐶, 𝐷⟩})) | |
2 | bj-fvmptunsn.nel | . . 3 ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴) | |
3 | eqid 2727 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
4 | 3 | dmmptss 6239 | . . . 4 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐴 |
5 | 4 | sseli 3974 | . . 3 ⊢ (𝐶 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐵) → 𝐶 ∈ 𝐴) |
6 | 2, 5 | nsyl 140 | . 2 ⊢ (𝜑 → ¬ 𝐶 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐵)) |
7 | bj-fvmptunsn1.ex1 | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
8 | bj-fvmptunsn1.ex2 | . 2 ⊢ (𝜑 → 𝐷 ∈ 𝑊) | |
9 | 1, 6, 7, 8 | bj-fununsn2 36669 | 1 ⊢ (𝜑 → (𝐹‘𝐶) = 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1534 ∈ wcel 2099 ∪ cun 3942 {csn 4624 ⟨cop 4630 ↦ cmpt 5225 dom cdm 5672 ‘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-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 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: bj-iomnnom 36674 |
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