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| Mirrors > Home > MPE Home > Th. List > updjudhf | Structured version Visualization version GIF version | ||
| Description: The mapping of an element of the disjoint union to the value of the corresponding function is a function. (Contributed by AV, 26-Jun-2022.) | 
| Ref | Expression | 
|---|---|
| updjud.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) | 
| updjud.g | ⊢ (𝜑 → 𝐺:𝐵⟶𝐶) | 
| updjudhf.h | ⊢ 𝐻 = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) | 
| Ref | Expression | 
|---|---|
| updjudhf | ⊢ (𝜑 → 𝐻:(𝐴 ⊔ 𝐵)⟶𝐶) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eldju2ndl 9964 | . . . . . 6 ⊢ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ∧ (1st ‘𝑥) = ∅) → (2nd ‘𝑥) ∈ 𝐴) | |
| 2 | 1 | ex 412 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ⊔ 𝐵) → ((1st ‘𝑥) = ∅ → (2nd ‘𝑥) ∈ 𝐴)) | 
| 3 | updjud.f | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) | |
| 4 | ffvelcdm 7101 | . . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐶 ∧ (2nd ‘𝑥) ∈ 𝐴) → (𝐹‘(2nd ‘𝑥)) ∈ 𝐶) | |
| 5 | 4 | ex 412 | . . . . . 6 ⊢ (𝐹:𝐴⟶𝐶 → ((2nd ‘𝑥) ∈ 𝐴 → (𝐹‘(2nd ‘𝑥)) ∈ 𝐶)) | 
| 6 | 3, 5 | syl 17 | . . . . 5 ⊢ (𝜑 → ((2nd ‘𝑥) ∈ 𝐴 → (𝐹‘(2nd ‘𝑥)) ∈ 𝐶)) | 
| 7 | 2, 6 | sylan9r 508 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 𝐵)) → ((1st ‘𝑥) = ∅ → (𝐹‘(2nd ‘𝑥)) ∈ 𝐶)) | 
| 8 | 7 | imp 406 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 𝐵)) ∧ (1st ‘𝑥) = ∅) → (𝐹‘(2nd ‘𝑥)) ∈ 𝐶) | 
| 9 | df-ne 2941 | . . . . 5 ⊢ ((1st ‘𝑥) ≠ ∅ ↔ ¬ (1st ‘𝑥) = ∅) | |
| 10 | eldju2ndr 9965 | . . . . . . 7 ⊢ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ∧ (1st ‘𝑥) ≠ ∅) → (2nd ‘𝑥) ∈ 𝐵) | |
| 11 | 10 | ex 412 | . . . . . 6 ⊢ (𝑥 ∈ (𝐴 ⊔ 𝐵) → ((1st ‘𝑥) ≠ ∅ → (2nd ‘𝑥) ∈ 𝐵)) | 
| 12 | updjud.g | . . . . . . 7 ⊢ (𝜑 → 𝐺:𝐵⟶𝐶) | |
| 13 | ffvelcdm 7101 | . . . . . . . 8 ⊢ ((𝐺:𝐵⟶𝐶 ∧ (2nd ‘𝑥) ∈ 𝐵) → (𝐺‘(2nd ‘𝑥)) ∈ 𝐶) | |
| 14 | 13 | ex 412 | . . . . . . 7 ⊢ (𝐺:𝐵⟶𝐶 → ((2nd ‘𝑥) ∈ 𝐵 → (𝐺‘(2nd ‘𝑥)) ∈ 𝐶)) | 
| 15 | 12, 14 | syl 17 | . . . . . 6 ⊢ (𝜑 → ((2nd ‘𝑥) ∈ 𝐵 → (𝐺‘(2nd ‘𝑥)) ∈ 𝐶)) | 
| 16 | 11, 15 | sylan9r 508 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 𝐵)) → ((1st ‘𝑥) ≠ ∅ → (𝐺‘(2nd ‘𝑥)) ∈ 𝐶)) | 
| 17 | 9, 16 | biimtrrid 243 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 𝐵)) → (¬ (1st ‘𝑥) = ∅ → (𝐺‘(2nd ‘𝑥)) ∈ 𝐶)) | 
| 18 | 17 | imp 406 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 𝐵)) ∧ ¬ (1st ‘𝑥) = ∅) → (𝐺‘(2nd ‘𝑥)) ∈ 𝐶) | 
| 19 | 8, 18 | ifclda 4561 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 𝐵)) → if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))) ∈ 𝐶) | 
| 20 | updjudhf.h | . 2 ⊢ 𝐻 = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥)))) | |
| 21 | 19, 20 | fmptd 7134 | 1 ⊢ (𝜑 → 𝐻:(𝐴 ⊔ 𝐵)⟶𝐶) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∅c0 4333 ifcif 4525 ↦ cmpt 5225 ⟶wf 6557 ‘cfv 6561 1st c1st 8012 2nd c2nd 8013 ⊔ cdju 9938 | 
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| 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-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 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-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-1st 8014 df-2nd 8015 df-1o 8506 df-dju 9941 | 
| This theorem is referenced by: updjudhcoinlf 9972 updjudhcoinrg 9973 updjud 9974 | 
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