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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 5250 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
3 | 1, 2 | eqtri 2768 | . 2 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} |
4 | 3 | fvopab4ndm 7059 | 1 ⊢ (¬ 𝑋 ∈ 𝐴 → (𝐹‘𝑋) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ∅c0 4352 {copab 5228 ↦ cmpt 5249 ‘cfv 6573 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-dm 5710 df-iota 6525 df-fv 6581 |
This theorem is referenced by: bropfvvvvlem 8132 bropfvvvv 8133 curry1val 8146 curry2val 8150 homarcl 18095 arwval 18110 coafval 18131 pcofval 25062 newval 27912 leftval 27920 rightval 27921 fvmptrab 47207 fpprbasnn 47603 setrec2mpt 48789 |
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