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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fnunres1 | Structured version Visualization version GIF version |
Description: Restriction of a disjoint union to the domain of the first function. (Contributed by Thierry Arnoux, 9-Dec-2021.) |
Ref | Expression |
---|---|
fnunres1 | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ∪ 𝐺) ↾ 𝐴) = 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fndm 6425 | . . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
2 | 1 | 3ad2ant1 1130 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → dom 𝐹 = 𝐴) |
3 | 2 | reseq2d 5818 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ∪ 𝐺) ↾ dom 𝐹) = ((𝐹 ∪ 𝐺) ↾ 𝐴)) |
4 | fnrel 6424 | . . . 4 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹) | |
5 | 4 | 3ad2ant1 1130 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → Rel 𝐹) |
6 | fndm 6425 | . . . . . 6 ⊢ (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵) | |
7 | 6 | 3ad2ant2 1131 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → dom 𝐺 = 𝐵) |
8 | 2, 7 | ineq12d 4140 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → (dom 𝐹 ∩ dom 𝐺) = (𝐴 ∩ 𝐵)) |
9 | simp3 1135 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∩ 𝐵) = ∅) | |
10 | 8, 9 | eqtrd 2833 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → (dom 𝐹 ∩ dom 𝐺) = ∅) |
11 | funresdm1 30368 | . . 3 ⊢ ((Rel 𝐹 ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹 ∪ 𝐺) ↾ dom 𝐹) = 𝐹) | |
12 | 5, 10, 11 | syl2anc 587 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ∪ 𝐺) ↾ dom 𝐹) = 𝐹) |
13 | 3, 12 | eqtr3d 2835 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ∪ 𝐺) ↾ 𝐴) = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1538 ∪ cun 3879 ∩ cin 3880 ∅c0 4243 dom cdm 5519 ↾ cres 5521 Rel wrel 5524 Fn wfn 6319 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-dm 5529 df-res 5531 df-fun 6326 df-fn 6327 |
This theorem is referenced by: fnunres2 30441 tocycfvres2 30803 cycpmconjslem2 30847 lbsdiflsp0 31110 actfunsnf1o 31985 |
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