Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
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 6678 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ V) | |
5 | 3, 4 | eqeltrrd 2911 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ V) |
6 | snidg 4589 | . . . . . 6 ⊢ (𝐵 ∈ V → 𝐵 ∈ {𝐵}) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ {𝐵}) |
8 | 3, 7 | eqeltrd 2910 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ {𝐵}) |
9 | 2, 8 | ralrimia 41274 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ {𝐵}) |
10 | nfcv 2974 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
11 | nfcv 2974 | . . . 4 ⊢ Ⅎ𝑥{𝐵} | |
12 | fconst7.x | . . . 4 ⊢ Ⅎ𝑥𝐹 | |
13 | 10, 11, 12 | ffnfvf 6875 | . . 3 ⊢ (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ {𝐵})) |
14 | 1, 9, 13 | sylanbrc 583 | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶{𝐵}) |
15 | fconst7.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
16 | fconst2g 6957 | . . 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 233 | 1 ⊢ (𝜑 → 𝐹 = (𝐴 × {𝐵})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 Ⅎwnf 1775 ∈ wcel 2105 Ⅎwnfc 2958 ∀wral 3135 Vcvv 3492 {csn 4557 × cxp 5546 Fn wfn 6343 ⟶wf 6344 ‘cfv 6348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 |
This theorem is referenced by: xlimconst 41982 |
Copyright terms: Public domain | W3C validator |