Theorem updjudhf 9821

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.)

Hypotheses

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))))

Assertion

Ref	Expression
updjudhf	⊢ (𝜑 → 𝐻:(𝐴 ⊔ 𝐵)⟶𝐶)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝜑,𝑥

Allowed substitution hints:   𝐹(𝑥)   𝐺(𝑥)   𝐻(𝑥)

Proof of Theorem updjudhf

Step	Hyp	Ref	Expression
1		eldju2ndl 9814	. . . . . 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 7014	. . . . . . 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 2929	. . . . 5 ⊢ ((1st ‘𝑥) ≠ ∅ ↔ ¬ (1st ‘𝑥) = ∅)
10		eldju2ndr 9815	. . . . . . 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 7014	. . . . . . . 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 4511	. 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 7047	1 ⊢ (𝜑 → 𝐻:(𝐴 ⊔ 𝐵)⟶𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∅c0 4283  ifcif 4475   ↦ cmpt 5172  ⟶wf 6477  ‘cfv 6481  1^st c1st 7919  2^nd c2nd 7920   ⊔ cdju 9788

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-1st 7921  df-2nd 7922  df-1o 8385  df-dju 9791

This theorem is referenced by:  updjudhcoinlf  9822  updjudhcoinrg  9823  updjud  9824