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Mirrors > Home > MPE Home > Th. List > Mathboxes > fvconstdomi | Structured version Visualization version GIF version |
Description: A constant function's value is dominated by the constant. (An artifact of our function value definition.) (Contributed by Zhi Wang, 18-Sep-2024.) |
Ref | Expression |
---|---|
fvconstdomi.1 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
fvconstdomi | ⊢ ((𝐴 × {𝐵})‘𝑋) ≼ 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmxpss 6193 | . . . . 5 ⊢ dom (𝐴 × {𝐵}) ⊆ 𝐴 | |
2 | 1 | sseli 3991 | . . . 4 ⊢ (𝑋 ∈ dom (𝐴 × {𝐵}) → 𝑋 ∈ 𝐴) |
3 | fvconstdomi.1 | . . . . 5 ⊢ 𝐵 ∈ V | |
4 | 3 | fvconst2 7224 | . . . 4 ⊢ (𝑋 ∈ 𝐴 → ((𝐴 × {𝐵})‘𝑋) = 𝐵) |
5 | 2, 4 | syl 17 | . . 3 ⊢ (𝑋 ∈ dom (𝐴 × {𝐵}) → ((𝐴 × {𝐵})‘𝑋) = 𝐵) |
6 | domrefg 9026 | . . . 4 ⊢ (𝐵 ∈ V → 𝐵 ≼ 𝐵) | |
7 | 3, 6 | ax-mp 5 | . . 3 ⊢ 𝐵 ≼ 𝐵 |
8 | 5, 7 | eqbrtrdi 5187 | . 2 ⊢ (𝑋 ∈ dom (𝐴 × {𝐵}) → ((𝐴 × {𝐵})‘𝑋) ≼ 𝐵) |
9 | ndmfv 6942 | . . 3 ⊢ (¬ 𝑋 ∈ dom (𝐴 × {𝐵}) → ((𝐴 × {𝐵})‘𝑋) = ∅) | |
10 | 3 | 0dom 9145 | . . 3 ⊢ ∅ ≼ 𝐵 |
11 | 9, 10 | eqbrtrdi 5187 | . 2 ⊢ (¬ 𝑋 ∈ dom (𝐴 × {𝐵}) → ((𝐴 × {𝐵})‘𝑋) ≼ 𝐵) |
12 | 8, 11 | pm2.61i 182 | 1 ⊢ ((𝐴 × {𝐵})‘𝑋) ≼ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 {csn 4631 class class class wbr 5148 × cxp 5687 dom cdm 5689 ‘cfv 6563 ≼ cdom 8982 |
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-pow 5371 ax-pr 5438 ax-un 7754 |
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-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-en 8985 df-dom 8986 |
This theorem is referenced by: f1omoALT 48692 |
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