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Mirrors > Home > MPE Home > Th. List > fvmptndm | Structured version Visualization version GIF version |
Description: Value of a function given by the maps-to notation, outside of its domain. (Contributed by AV, 31-Dec-2020.) |
Ref | Expression |
---|---|
fvmptndm.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
Ref | Expression |
---|---|
fvmptndm | ⊢ (¬ 𝑋 ∈ 𝐴 → (𝐹‘𝑋) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvmptndm.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | df-mpt 5232 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
3 | 1, 2 | eqtri 2763 | . 2 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} |
4 | 3 | fvopab4ndm 7046 | 1 ⊢ (¬ 𝑋 ∈ 𝐴 → (𝐹‘𝑋) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∅c0 4339 {copab 5210 ↦ cmpt 5231 ‘cfv 6563 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-dm 5699 df-iota 6516 df-fv 6571 |
This theorem is referenced by: bropfvvvvlem 8115 bropfvvvv 8116 curry1val 8129 curry2val 8133 homarcl 18082 arwval 18097 coafval 18118 pcofval 25057 newval 27909 leftval 27917 rightval 27918 fvmptrab 47242 fpprbasnn 47654 setrec2mpt 48928 |
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