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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 9921 | . . . . . 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 7077 | . . . . . . 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 2935 | . . . . 5 ⊢ ((1st ‘𝑥) ≠ ∅ ↔ ¬ (1st ‘𝑥) = ∅) | |
10 | eldju2ndr 9922 | . . . . . . 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 7077 | . . . . . . . 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 242 | . . . 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 4558 | . 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 7109 | 1 ⊢ (𝜑 → 𝐻:(𝐴 ⊔ 𝐵)⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ∅c0 4317 ifcif 4523 ↦ cmpt 5224 ⟶wf 6533 ‘cfv 6537 1st c1st 7972 2nd c2nd 7973 ⊔ cdju 9895 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-fv 6545 df-1st 7974 df-2nd 7975 df-1o 8467 df-dju 9898 |
This theorem is referenced by: updjudhcoinlf 9929 updjudhcoinrg 9930 updjud 9931 |
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