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 6455 | . . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
2 | 1 | 3ad2ant1 1129 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → dom 𝐹 = 𝐴) |
3 | 2 | reseq2d 5853 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ∪ 𝐺) ↾ dom 𝐹) = ((𝐹 ∪ 𝐺) ↾ 𝐴)) |
4 | fnrel 6454 | . . . 4 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹) | |
5 | 4 | 3ad2ant1 1129 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → Rel 𝐹) |
6 | fndm 6455 | . . . . . 6 ⊢ (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵) | |
7 | 6 | 3ad2ant2 1130 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → dom 𝐺 = 𝐵) |
8 | 2, 7 | ineq12d 4190 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → (dom 𝐹 ∩ dom 𝐺) = (𝐴 ∩ 𝐵)) |
9 | simp3 1134 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∩ 𝐵) = ∅) | |
10 | 8, 9 | eqtrd 2856 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → (dom 𝐹 ∩ dom 𝐺) = ∅) |
11 | funresdm1 30355 | . . 3 ⊢ ((Rel 𝐹 ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹 ∪ 𝐺) ↾ dom 𝐹) = 𝐹) | |
12 | 5, 10, 11 | syl2anc 586 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ∪ 𝐺) ↾ dom 𝐹) = 𝐹) |
13 | 3, 12 | eqtr3d 2858 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ∪ 𝐺) ↾ 𝐴) = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∪ cun 3934 ∩ cin 3935 ∅c0 4291 dom cdm 5555 ↾ cres 5557 Rel wrel 5560 Fn wfn 6350 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 df-dm 5565 df-res 5567 df-fun 6357 df-fn 6358 |
This theorem is referenced by: fnunres2 30424 tocycfvres2 30753 cycpmconjslem2 30797 lbsdiflsp0 31022 actfunsnf1o 31875 |
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