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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 6160 | . . . . . 6 ⊢ dom (𝐴 × {𝐵}) ⊆ 𝐴 | |
3 | 2 | sseli 3970 | . . . . 5 ⊢ (𝑋 ∈ dom (𝐴 × {𝐵}) → 𝑋 ∈ 𝐴) |
4 | fvconst0ci.1 | . . . . . 6 ⊢ 𝐵 ∈ V | |
5 | 4 | fvconst2 7197 | . . . . 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 2776 | . . 3 ⊢ (𝑋 ∈ dom (𝐴 × {𝐵}) → 𝑌 = 𝐵) |
8 | 7 | olcd 871 | . 2 ⊢ (𝑋 ∈ dom (𝐴 × {𝐵}) → (𝑌 = ∅ ∨ 𝑌 = 𝐵)) |
9 | ndmfv 6916 | . . . 4 ⊢ (¬ 𝑋 ∈ dom (𝐴 × {𝐵}) → ((𝐴 × {𝐵})‘𝑋) = ∅) | |
10 | 1, 9 | eqtrid 2776 | . . 3 ⊢ (¬ 𝑋 ∈ dom (𝐴 × {𝐵}) → 𝑌 = ∅) |
11 | 10 | orcd 870 | . 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 844 = wceq 1533 ∈ wcel 2098 Vcvv 3466 ∅c0 4314 {csn 4620 × cxp 5664 dom cdm 5666 ‘cfv 6533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-fv 6541 |
This theorem is referenced by: f1omo 47715 |
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