|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| 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 6670 | . . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
| 2 | 1 | 3ad2ant1 1133 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → dom 𝐹 = 𝐴) | 
| 3 | 2 | reseq2d 5996 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ∪ 𝐺) ↾ dom 𝐹) = ((𝐹 ∪ 𝐺) ↾ 𝐴)) | 
| 4 | fnrel 6669 | . . . 4 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹) | |
| 5 | 4 | 3ad2ant1 1133 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → Rel 𝐹) | 
| 6 | fndm 6670 | . . . . . 6 ⊢ (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵) | |
| 7 | 6 | 3ad2ant2 1134 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → dom 𝐺 = 𝐵) | 
| 8 | 2, 7 | ineq12d 4220 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → (dom 𝐹 ∩ dom 𝐺) = (𝐴 ∩ 𝐵)) | 
| 9 | simp3 1138 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∩ 𝐵) = ∅) | |
| 10 | 8, 9 | eqtrd 2776 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → (dom 𝐹 ∩ dom 𝐺) = ∅) | 
| 11 | relresdm1 6050 | . . 3 ⊢ ((Rel 𝐹 ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹 ∪ 𝐺) ↾ dom 𝐹) = 𝐹) | |
| 12 | 5, 10, 11 | syl2anc 584 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ∪ 𝐺) ↾ dom 𝐹) = 𝐹) | 
| 13 | 3, 12 | eqtr3d 2778 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ∪ 𝐺) ↾ 𝐴) = 𝐹) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1539 ∪ cun 3948 ∩ cin 3949 ∅c0 4332 dom cdm 5684 ↾ cres 5686 Rel wrel 5689 Fn wfn 6555 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 df-rel 5691 df-dm 5694 df-res 5696 df-fun 6562 df-fn 6563 | 
| This theorem is referenced by: fnunres2 6680 tocycfvres2 33132 cycpmconjslem2 33176 lbsdiflsp0 33678 actfunsnf1o 34620 dvun 42394 evlselvlem 42601 evlselv 42602 | 
| Copyright terms: Public domain | W3C validator |