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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 6013 | . . . . 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 6988 | . . . 4 ⊢ (𝑋 ∈ 𝐴 → ((𝐴 × {𝐵})‘𝑋) = 𝐵) |
5 | 2, 4 | syl 17 | . . 3 ⊢ (𝑋 ∈ dom (𝐴 × {𝐵}) → ((𝐴 × {𝐵})‘𝑋) = 𝐵) |
6 | domrefg 8602 | . . . 4 ⊢ (𝐵 ∈ V → 𝐵 ≼ 𝐵) | |
7 | 3, 6 | ax-mp 5 | . . 3 ⊢ 𝐵 ≼ 𝐵 |
8 | 5, 7 | eqbrtrdi 5079 | . 2 ⊢ (𝑋 ∈ dom (𝐴 × {𝐵}) → ((𝐴 × {𝐵})‘𝑋) ≼ 𝐵) |
9 | ndmfv 6716 | . . 3 ⊢ (¬ 𝑋 ∈ dom (𝐴 × {𝐵}) → ((𝐴 × {𝐵})‘𝑋) = ∅) | |
10 | 3 | 0dom 8709 | . . 3 ⊢ ∅ ≼ 𝐵 |
11 | 9, 10 | eqbrtrdi 5079 | . 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 1542 ∈ wcel 2114 Vcvv 3400 ∅c0 4221 {csn 4526 class class class wbr 5040 × cxp 5533 dom cdm 5535 ‘cfv 6349 ≼ cdom 8565 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3402 df-sbc 3686 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-en 8568 df-dom 8569 |
This theorem is referenced by: f1omoALT 45759 |
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