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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 2820 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
4 | 3 | dmmptss 6088 | . . . 4 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐴 |
5 | 4 | sseli 3956 | . . 3 ⊢ (𝐶 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐵) → 𝐶 ∈ 𝐴) |
6 | 2, 5 | nsyl 142 | . 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 34558 | 1 ⊢ (𝜑 → (𝐹‘𝐶) = 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1536 ∈ wcel 2113 ∪ cun 3927 {csn 4560 〈cop 4566 ↦ cmpt 5139 dom cdm 5548 ‘cfv 6348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fv 6356 |
This theorem is referenced by: bj-iomnnom 34563 |
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