Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-fununsn1 | Structured version Visualization version GIF version |
Description: Value of a function expressed as a union of a function and a singleton on a couple (with disjoint domain) at a point not equal to the first component of that couple. (Contributed by BJ, 18-Mar-2023.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-fununsn.un | ⊢ (𝜑 → 𝐹 = (𝐺 ∪ {〈𝐵, 𝐶〉})) |
bj-fununsn1.neq | ⊢ (𝜑 → ¬ 𝐴 = 𝐵) |
Ref | Expression |
---|---|
bj-fununsn1 | ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-fununsn.un | . 2 ⊢ (𝜑 → 𝐹 = (𝐺 ∪ {〈𝐵, 𝐶〉})) | |
2 | dmsnopss 6073 | . . . 4 ⊢ dom {〈𝐵, 𝐶〉} ⊆ {𝐵} | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → dom {〈𝐵, 𝐶〉} ⊆ {𝐵}) |
4 | bj-fununsn1.neq | . . . 4 ⊢ (𝜑 → ¬ 𝐴 = 𝐵) | |
5 | elsni 4586 | . . . 4 ⊢ (𝐴 ∈ {𝐵} → 𝐴 = 𝐵) | |
6 | 4, 5 | nsyl 142 | . . 3 ⊢ (𝜑 → ¬ 𝐴 ∈ {𝐵}) |
7 | 3, 6 | ssneldd 3972 | . 2 ⊢ (𝜑 → ¬ 𝐴 ∈ dom {〈𝐵, 𝐶〉}) |
8 | 1, 7 | bj-funun 34536 | 1 ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1537 ∈ wcel 2114 ∪ cun 3936 ⊆ wss 3938 {csn 4569 〈cop 4575 dom cdm 5557 ‘cfv 6357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-xp 5563 df-cnv 5565 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fv 6365 |
This theorem is referenced by: bj-fvsnun1 34539 bj-fvmptunsn2 34542 |
Copyright terms: Public domain | W3C validator |