![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > opfv | Structured version Visualization version GIF version |
Description: Value of a function producing ordered pairs. (Contributed by Thierry Arnoux, 3-Jan-2017.) |
Ref | Expression |
---|---|
opfv | ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) = 〈((1st ∘ 𝐹)‘𝑥), ((2nd ∘ 𝐹)‘𝑥)〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplr 768 | . . . 4 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → ran 𝐹 ⊆ (V × V)) | |
2 | fvelrn 7081 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹) | |
3 | 2 | adantlr 714 | . . . 4 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹) |
4 | 1, 3 | sseldd 3980 | . . 3 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ (V × V)) |
5 | 1st2ndb 8028 | . . 3 ⊢ ((𝐹‘𝑥) ∈ (V × V) ↔ (𝐹‘𝑥) = 〈(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))〉) | |
6 | 4, 5 | sylib 217 | . 2 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) = 〈(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))〉) |
7 | fvco 6991 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((1st ∘ 𝐹)‘𝑥) = (1st ‘(𝐹‘𝑥))) | |
8 | fvco 6991 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((2nd ∘ 𝐹)‘𝑥) = (2nd ‘(𝐹‘𝑥))) | |
9 | 7, 8 | opeq12d 4878 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → 〈((1st ∘ 𝐹)‘𝑥), ((2nd ∘ 𝐹)‘𝑥)〉 = 〈(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))〉) |
10 | 9 | adantlr 714 | . 2 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → 〈((1st ∘ 𝐹)‘𝑥), ((2nd ∘ 𝐹)‘𝑥)〉 = 〈(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))〉) |
11 | 6, 10 | eqtr4d 2771 | 1 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) = 〈((1st ∘ 𝐹)‘𝑥), ((2nd ∘ 𝐹)‘𝑥)〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 Vcvv 3470 ⊆ wss 3945 〈cop 4631 × cxp 5671 dom cdm 5673 ran crn 5674 ∘ ccom 5677 Fun wfun 6537 ‘cfv 6543 1st c1st 7986 2nd c2nd 7987 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5294 ax-nul 5301 ax-pr 5424 ax-un 7735 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6495 df-fun 6545 df-fn 6546 df-fv 6551 df-1st 7988 df-2nd 7989 |
This theorem is referenced by: xppreima 32426 xppreima2 32431 |
Copyright terms: Public domain | W3C validator |