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Mirrors > Home > MPE Home > Th. List > 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 6662 | . . . 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 5989 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ∪ 𝐺) ↾ dom 𝐹) = ((𝐹 ∪ 𝐺) ↾ 𝐴)) |
4 | fnrel 6661 | . . . 4 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹) | |
5 | 4 | 3ad2ant1 1130 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → Rel 𝐹) |
6 | fndm 6662 | . . . . . 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 4215 | . . . 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 2768 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → (dom 𝐹 ∩ dom 𝐺) = ∅) |
11 | relresdm1 6042 | . . 3 ⊢ ((Rel 𝐹 ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹 ∪ 𝐺) ↾ dom 𝐹) = 𝐹) | |
12 | 5, 10, 11 | syl2anc 582 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ∪ 𝐺) ↾ dom 𝐹) = 𝐹) |
13 | 3, 12 | eqtr3d 2770 | 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 1533 ∪ cun 3947 ∩ cin 3948 ∅c0 4326 dom cdm 5682 ↾ cres 5684 Rel wrel 5687 Fn wfn 6548 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-br 5153 df-opab 5215 df-xp 5688 df-rel 5689 df-dm 5692 df-res 5694 df-fun 6555 df-fn 6556 |
This theorem is referenced by: fnunres2 6672 tocycfvres2 32861 cycpmconjslem2 32905 lbsdiflsp0 33365 actfunsnf1o 34277 evlselvlem 41868 evlselv 41869 |
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