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| Description: Restriction of a disjoint union to the domain of the second function. (Contributed by Thierry Arnoux, 12-Oct-2023.) | 
| Ref | Expression | 
|---|---|
| fnunres2 | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ∪ 𝐺) ↾ 𝐵) = 𝐺) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | uncom 4158 | . . 3 ⊢ (𝐹 ∪ 𝐺) = (𝐺 ∪ 𝐹) | |
| 2 | 1 | reseq1i 5993 | . 2 ⊢ ((𝐹 ∪ 𝐺) ↾ 𝐵) = ((𝐺 ∪ 𝐹) ↾ 𝐵) | 
| 3 | ineqcom 4210 | . . . 4 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐵 ∩ 𝐴) = ∅) | |
| 4 | fnunres1 6680 | . . . 4 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐹 Fn 𝐴 ∧ (𝐵 ∩ 𝐴) = ∅) → ((𝐺 ∪ 𝐹) ↾ 𝐵) = 𝐺) | |
| 5 | 3, 4 | syl3an3b 1407 | . . 3 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐹 Fn 𝐴 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐺 ∪ 𝐹) ↾ 𝐵) = 𝐺) | 
| 6 | 5 | 3com12 1124 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐺 ∪ 𝐹) ↾ 𝐵) = 𝐺) | 
| 7 | 2, 6 | eqtrid 2789 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ∪ 𝐺) ↾ 𝐵) = 𝐺) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∪ cun 3949 ∩ cin 3950 ∅c0 4333 ↾ cres 5687 Fn wfn 6556 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-dm 5695 df-res 5697 df-fun 6563 df-fn 6564 | 
| This theorem is referenced by: tocycfvres1 33130 cycpmconjslem2 33175 dvun 42389 evlselvlem 42596 evlselv 42597 | 
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