Mathbox for Stefan O'Rear |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > constmap | Structured version Visualization version GIF version |
Description: A constant (represented
without dummy variables) is an element of a
function set.
_Note: In the following development, we will be quite often quantifying over functions and points in N-dimensional space (which are equivalent to functions from an "index set"). Many of the following theorems exist to transfer standard facts about functions to elements of function sets._ (Contributed by Stefan O'Rear, 30-Aug-2014.) (Revised by Stefan O'Rear, 5-May-2015.) |
Ref | Expression |
---|---|
constmap.1 | ⊢ 𝐴 ∈ V |
constmap.3 | ⊢ 𝐶 ∈ V |
Ref | Expression |
---|---|
constmap | ⊢ (𝐵 ∈ 𝐶 → (𝐴 × {𝐵}) ∈ (𝐶 ↑𝑚 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fconst6g 6094 | . 2 ⊢ (𝐵 ∈ 𝐶 → (𝐴 × {𝐵}):𝐴⟶𝐶) | |
2 | constmap.3 | . . 3 ⊢ 𝐶 ∈ V | |
3 | constmap.1 | . . 3 ⊢ 𝐴 ∈ V | |
4 | 2, 3 | elmap 7886 | . 2 ⊢ ((𝐴 × {𝐵}) ∈ (𝐶 ↑𝑚 𝐴) ↔ (𝐴 × {𝐵}):𝐴⟶𝐶) |
5 | 1, 4 | sylibr 224 | 1 ⊢ (𝐵 ∈ 𝐶 → (𝐴 × {𝐵}) ∈ (𝐶 ↑𝑚 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 Vcvv 3200 {csn 4177 × cxp 5112 ⟶wf 5884 (class class class)co 6650 ↑𝑚 cmap 7857 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-map 7859 |
This theorem is referenced by: mzpclall 37290 mzpindd 37309 |
Copyright terms: Public domain | W3C validator |