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Mathbox for Thierry Arnoux |
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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 752 | . . . 4 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → ran 𝐹 ⊆ (V × V)) | |
2 | fvelrn 6493 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹) | |
3 | 2 | adantlr 694 | . . . 4 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹) |
4 | 1, 3 | sseldd 3753 | . . 3 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ (V × V)) |
5 | 1st2ndb 7353 | . . 3 ⊢ ((𝐹‘𝑥) ∈ (V × V) ↔ (𝐹‘𝑥) = 〈(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))〉) | |
6 | 4, 5 | sylib 208 | . 2 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) = 〈(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))〉) |
7 | fvco 6414 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((1st ∘ 𝐹)‘𝑥) = (1st ‘(𝐹‘𝑥))) | |
8 | fvco 6414 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((2nd ∘ 𝐹)‘𝑥) = (2nd ‘(𝐹‘𝑥))) | |
9 | 7, 8 | opeq12d 4547 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → 〈((1st ∘ 𝐹)‘𝑥), ((2nd ∘ 𝐹)‘𝑥)〉 = 〈(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))〉) |
10 | 9 | adantlr 694 | . 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 2808 | 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 382 = wceq 1631 ∈ wcel 2145 Vcvv 3351 ⊆ wss 3723 〈cop 4322 × cxp 5247 dom cdm 5249 ran crn 5250 ∘ ccom 5253 Fun wfun 6023 ‘cfv 6029 1st c1st 7311 2nd c2nd 7312 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7094 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5992 df-fun 6031 df-fn 6032 df-fv 6037 df-1st 7313 df-2nd 7314 |
This theorem is referenced by: xppreima 29782 xppreima2 29783 |
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