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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 9085 | . . . . . 6 ⊢ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ∧ (1st ‘𝑥) = ∅) → (2nd ‘𝑥) ∈ 𝐴) | |
2 | 1 | ex 403 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ⊔ 𝐵) → ((1st ‘𝑥) = ∅ → (2nd ‘𝑥) ∈ 𝐴)) |
3 | updjud.f | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) | |
4 | ffvelrn 6623 | . . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐶 ∧ (2nd ‘𝑥) ∈ 𝐴) → (𝐹‘(2nd ‘𝑥)) ∈ 𝐶) | |
5 | 4 | ex 403 | . . . . . 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 504 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 𝐵)) → ((1st ‘𝑥) = ∅ → (𝐹‘(2nd ‘𝑥)) ∈ 𝐶)) |
8 | 7 | imp 397 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 𝐵)) ∧ (1st ‘𝑥) = ∅) → (𝐹‘(2nd ‘𝑥)) ∈ 𝐶) |
9 | df-ne 2970 | . . . . 5 ⊢ ((1st ‘𝑥) ≠ ∅ ↔ ¬ (1st ‘𝑥) = ∅) | |
10 | eldju2ndr 9086 | . . . . . . 7 ⊢ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ∧ (1st ‘𝑥) ≠ ∅) → (2nd ‘𝑥) ∈ 𝐵) | |
11 | 10 | ex 403 | . . . . . 6 ⊢ (𝑥 ∈ (𝐴 ⊔ 𝐵) → ((1st ‘𝑥) ≠ ∅ → (2nd ‘𝑥) ∈ 𝐵)) |
12 | updjud.g | . . . . . . 7 ⊢ (𝜑 → 𝐺:𝐵⟶𝐶) | |
13 | ffvelrn 6623 | . . . . . . . 8 ⊢ ((𝐺:𝐵⟶𝐶 ∧ (2nd ‘𝑥) ∈ 𝐵) → (𝐺‘(2nd ‘𝑥)) ∈ 𝐶) | |
14 | 13 | ex 403 | . . . . . . 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 504 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 𝐵)) → ((1st ‘𝑥) ≠ ∅ → (𝐺‘(2nd ‘𝑥)) ∈ 𝐶)) |
17 | 9, 16 | syl5bir 235 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 𝐵)) → (¬ (1st ‘𝑥) = ∅ → (𝐺‘(2nd ‘𝑥)) ∈ 𝐶)) |
18 | 17 | imp 397 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 𝐵)) ∧ ¬ (1st ‘𝑥) = ∅) → (𝐺‘(2nd ‘𝑥)) ∈ 𝐶) |
19 | 8, 18 | ifclda 4341 | . 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 6650 | 1 ⊢ (𝜑 → 𝐻:(𝐴 ⊔ 𝐵)⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ∅c0 4141 ifcif 4307 ↦ cmpt 4967 ⟶wf 6133 ‘cfv 6137 1st c1st 7445 2nd c2nd 7446 ⊔ cdju 9060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-fv 6145 df-1st 7447 df-2nd 7448 df-1o 7845 df-dju 9063 |
This theorem is referenced by: updjudhcoinlf 9093 updjudhcoinrg 9094 updjud 9095 |
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