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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 6906 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ V) | |
5 | 3, 4 | eqeltrrd 2833 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ V) |
6 | snidg 4662 | . . . . . 6 ⊢ (𝐵 ∈ V → 𝐵 ∈ {𝐵}) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ {𝐵}) |
8 | 3, 7 | eqeltrd 2832 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ {𝐵}) |
9 | 2, 8 | ralrimia 3254 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ {𝐵}) |
10 | nfcv 2902 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
11 | nfcv 2902 | . . . 4 ⊢ Ⅎ𝑥{𝐵} | |
12 | fconst7.x | . . . 4 ⊢ Ⅎ𝑥𝐹 | |
13 | 10, 11, 12 | ffnfvf 7121 | . . 3 ⊢ (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ {𝐵})) |
14 | 1, 9, 13 | sylanbrc 582 | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶{𝐵}) |
15 | fconst7.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
16 | fconst2g 7206 | . . 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 395 = wceq 1540 Ⅎwnf 1784 ∈ wcel 2105 Ⅎwnfc 2882 ∀wral 3060 Vcvv 3473 {csn 4628 × cxp 5674 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 |
This theorem is referenced by: xlimconst 45003 |
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