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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 6014 | . . . . 5 ⊢ dom (𝐴 × {𝐵}) ⊆ 𝐴 | |
2 | 1 | sseli 3883 | . . . 4 ⊢ (𝑋 ∈ dom (𝐴 × {𝐵}) → 𝑋 ∈ 𝐴) |
3 | fvconstdomi.1 | . . . . 5 ⊢ 𝐵 ∈ V | |
4 | 3 | fvconst2 6997 | . . . 4 ⊢ (𝑋 ∈ 𝐴 → ((𝐴 × {𝐵})‘𝑋) = 𝐵) |
5 | 2, 4 | syl 17 | . . 3 ⊢ (𝑋 ∈ dom (𝐴 × {𝐵}) → ((𝐴 × {𝐵})‘𝑋) = 𝐵) |
6 | domrefg 8641 | . . . 4 ⊢ (𝐵 ∈ V → 𝐵 ≼ 𝐵) | |
7 | 3, 6 | ax-mp 5 | . . 3 ⊢ 𝐵 ≼ 𝐵 |
8 | 5, 7 | eqbrtrdi 5078 | . 2 ⊢ (𝑋 ∈ dom (𝐴 × {𝐵}) → ((𝐴 × {𝐵})‘𝑋) ≼ 𝐵) |
9 | ndmfv 6725 | . . 3 ⊢ (¬ 𝑋 ∈ dom (𝐴 × {𝐵}) → ((𝐴 × {𝐵})‘𝑋) = ∅) | |
10 | 3 | 0dom 8754 | . . 3 ⊢ ∅ ≼ 𝐵 |
11 | 9, 10 | eqbrtrdi 5078 | . 2 ⊢ (¬ 𝑋 ∈ dom (𝐴 × {𝐵}) → ((𝐴 × {𝐵})‘𝑋) ≼ 𝐵) |
12 | 8, 11 | pm2.61i 185 | 1 ⊢ ((𝐴 × {𝐵})‘𝑋) ≼ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1543 ∈ wcel 2112 Vcvv 3398 ∅c0 4223 {csn 4527 class class class wbr 5039 × cxp 5534 dom cdm 5536 ‘cfv 6358 ≼ cdom 8602 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-en 8605 df-dom 8606 |
This theorem is referenced by: f1omoALT 45805 |
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