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Mirrors > Home > MPE Home > Th. List > Mathboxes > fvconst0ci | Structured version Visualization version GIF version |
Description: A constant function's value is either the constant or the empty set. (An artifact of our function value definition.) (Contributed by Zhi Wang, 18-Sep-2024.) |
Ref | Expression |
---|---|
fvconst0ci.1 | ⊢ 𝐵 ∈ V |
fvconst0ci.2 | ⊢ 𝑌 = ((𝐴 × {𝐵})‘𝑋) |
Ref | Expression |
---|---|
fvconst0ci | ⊢ (𝑌 = ∅ ∨ 𝑌 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvconst0ci.2 | . . . 4 ⊢ 𝑌 = ((𝐴 × {𝐵})‘𝑋) | |
2 | dmxpss 6162 | . . . . . 6 ⊢ dom (𝐴 × {𝐵}) ⊆ 𝐴 | |
3 | 2 | sseli 3976 | . . . . 5 ⊢ (𝑋 ∈ dom (𝐴 × {𝐵}) → 𝑋 ∈ 𝐴) |
4 | fvconst0ci.1 | . . . . . 6 ⊢ 𝐵 ∈ V | |
5 | 4 | fvconst2 7192 | . . . . 5 ⊢ (𝑋 ∈ 𝐴 → ((𝐴 × {𝐵})‘𝑋) = 𝐵) |
6 | 3, 5 | syl 17 | . . . 4 ⊢ (𝑋 ∈ dom (𝐴 × {𝐵}) → ((𝐴 × {𝐵})‘𝑋) = 𝐵) |
7 | 1, 6 | eqtrid 2785 | . . 3 ⊢ (𝑋 ∈ dom (𝐴 × {𝐵}) → 𝑌 = 𝐵) |
8 | 7 | olcd 873 | . 2 ⊢ (𝑋 ∈ dom (𝐴 × {𝐵}) → (𝑌 = ∅ ∨ 𝑌 = 𝐵)) |
9 | ndmfv 6916 | . . . 4 ⊢ (¬ 𝑋 ∈ dom (𝐴 × {𝐵}) → ((𝐴 × {𝐵})‘𝑋) = ∅) | |
10 | 1, 9 | eqtrid 2785 | . . 3 ⊢ (¬ 𝑋 ∈ dom (𝐴 × {𝐵}) → 𝑌 = ∅) |
11 | 10 | orcd 872 | . 2 ⊢ (¬ 𝑋 ∈ dom (𝐴 × {𝐵}) → (𝑌 = ∅ ∨ 𝑌 = 𝐵)) |
12 | 8, 11 | pm2.61i 182 | 1 ⊢ (𝑌 = ∅ ∨ 𝑌 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 846 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∅c0 4320 {csn 4624 × cxp 5670 dom cdm 5672 ‘cfv 6535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-fv 6543 |
This theorem is referenced by: f1omo 47367 |
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