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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fconst7 | Structured version Visualization version GIF version |
Description: An alternative way to express a constant function. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
Ref | Expression |
---|---|
fconst7.p | ⊢ Ⅎ𝑥𝜑 |
fconst7.x | ⊢ Ⅎ𝑥𝐹 |
fconst7.f | ⊢ (𝜑 → 𝐹 Fn 𝐴) |
fconst7.b | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
fconst7.e | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵) |
Ref | Expression |
---|---|
fconst7 | ⊢ (𝜑 → 𝐹 = (𝐴 × {𝐵})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fconst7.f | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) | |
2 | fconst7.p | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
3 | fconst7.e | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵) | |
4 | fvexd 6903 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ V) | |
5 | 3, 4 | eqeltrrd 2835 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ V) |
6 | snidg 4661 | . . . . . 6 ⊢ (𝐵 ∈ V → 𝐵 ∈ {𝐵}) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ {𝐵}) |
8 | 3, 7 | eqeltrd 2834 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ {𝐵}) |
9 | 2, 8 | ralrimia 3256 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ {𝐵}) |
10 | nfcv 2904 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
11 | nfcv 2904 | . . . 4 ⊢ Ⅎ𝑥{𝐵} | |
12 | fconst7.x | . . . 4 ⊢ Ⅎ𝑥𝐹 | |
13 | 10, 11, 12 | ffnfvf 7114 | . . 3 ⊢ (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ {𝐵})) |
14 | 1, 9, 13 | sylanbrc 584 | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶{𝐵}) |
15 | fconst7.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
16 | fconst2g 7199 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵}))) | |
17 | 15, 16 | syl 17 | . 2 ⊢ (𝜑 → (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵}))) |
18 | 14, 17 | mpbid 231 | 1 ⊢ (𝜑 → 𝐹 = (𝐴 × {𝐵})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 Ⅎwnf 1786 ∈ wcel 2107 Ⅎwnfc 2884 ∀wral 3062 Vcvv 3475 {csn 4627 × cxp 5673 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 |
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 5298 ax-nul 5305 ax-pr 5426 |
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-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-fv 6548 |
This theorem is referenced by: xlimconst 44476 |
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