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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 787 | . . . 4 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → ran 𝐹 ⊆ (V × V)) | |
2 | fvelrn 6602 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹) | |
3 | 2 | adantlr 708 | . . . 4 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹) |
4 | 1, 3 | sseldd 3829 | . . 3 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ (V × V)) |
5 | 1st2ndb 7469 | . . 3 ⊢ ((𝐹‘𝑥) ∈ (V × V) ↔ (𝐹‘𝑥) = 〈(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))〉) | |
6 | 4, 5 | sylib 210 | . 2 ⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) = 〈(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))〉) |
7 | fvco 6522 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((1st ∘ 𝐹)‘𝑥) = (1st ‘(𝐹‘𝑥))) | |
8 | fvco 6522 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((2nd ∘ 𝐹)‘𝑥) = (2nd ‘(𝐹‘𝑥))) | |
9 | 7, 8 | opeq12d 4632 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → 〈((1st ∘ 𝐹)‘𝑥), ((2nd ∘ 𝐹)‘𝑥)〉 = 〈(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))〉) |
10 | 9 | adantlr 708 | . 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 2865 | 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 386 = wceq 1658 ∈ wcel 2166 Vcvv 3415 ⊆ wss 3799 〈cop 4404 × cxp 5341 dom cdm 5343 ran crn 5344 ∘ ccom 5347 Fun wfun 6118 ‘cfv 6124 1st c1st 7427 2nd c2nd 7428 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-sbc 3664 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-opab 4937 df-mpt 4954 df-id 5251 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-iota 6087 df-fun 6126 df-fn 6127 df-fv 6132 df-1st 7429 df-2nd 7430 |
This theorem is referenced by: xppreima 29999 xppreima2 30000 |
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