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Mirrors > Home > MPE Home > Th. List > fvopab4ndm | Structured version Visualization version GIF version |
Description: Value of a function given by an ordered-pair class abstraction, outside of its domain. (Contributed by NM, 28-Mar-2008.) |
Ref | Expression |
---|---|
fvopab4ndm.1 | ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} |
Ref | Expression |
---|---|
fvopab4ndm | ⊢ (¬ 𝐵 ∈ 𝐴 → (𝐹‘𝐵) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvopab4ndm.1 | . . . . . 6 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
2 | 1 | dmeqi 5768 | . . . . 5 ⊢ dom 𝐹 = dom {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} |
3 | dmopabss 5782 | . . . . 5 ⊢ dom {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴 | |
4 | 2, 3 | eqsstri 4001 | . . . 4 ⊢ dom 𝐹 ⊆ 𝐴 |
5 | 4 | sseli 3963 | . . 3 ⊢ (𝐵 ∈ dom 𝐹 → 𝐵 ∈ 𝐴) |
6 | 5 | con3i 157 | . 2 ⊢ (¬ 𝐵 ∈ 𝐴 → ¬ 𝐵 ∈ dom 𝐹) |
7 | ndmfv 6695 | . 2 ⊢ (¬ 𝐵 ∈ dom 𝐹 → (𝐹‘𝐵) = ∅) | |
8 | 6, 7 | syl 17 | 1 ⊢ (¬ 𝐵 ∈ 𝐴 → (𝐹‘𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∅c0 4291 {copab 5121 dom cdm 5550 ‘cfv 6350 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-dm 5560 df-iota 6309 df-fv 6358 |
This theorem is referenced by: fvmptndm 6793 |
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