Theorem updjudhf 9344

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 9337	. . . . . 6 ⊢ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ∧ (1st ‘𝑥) = ∅) → (2nd ‘𝑥) ∈ 𝐴)
2	1	ex 416	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ⊔ 𝐵) → ((1st ‘𝑥) = ∅ → (2nd ‘𝑥) ∈ 𝐴))
3		updjud.f	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)
4		ffvelrn 6826	. . . . . . 7 ⊢ ((𝐹:𝐴⟶𝐶 ∧ (2nd ‘𝑥) ∈ 𝐴) → (𝐹‘(2nd ‘𝑥)) ∈ 𝐶)
5	4	ex 416	. . . . . 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 512	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 𝐵)) → ((1st ‘𝑥) = ∅ → (𝐹‘(2nd ‘𝑥)) ∈ 𝐶))
8	7	imp 410	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 𝐵)) ∧ (1st ‘𝑥) = ∅) → (𝐹‘(2nd ‘𝑥)) ∈ 𝐶)
9		df-ne 2988	. . . . 5 ⊢ ((1st ‘𝑥) ≠ ∅ ↔ ¬ (1st ‘𝑥) = ∅)
10		eldju2ndr 9338	. . . . . . 7 ⊢ ((𝑥 ∈ (𝐴 ⊔ 𝐵) ∧ (1st ‘𝑥) ≠ ∅) → (2nd ‘𝑥) ∈ 𝐵)
11	10	ex 416	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ⊔ 𝐵) → ((1st ‘𝑥) ≠ ∅ → (2nd ‘𝑥) ∈ 𝐵))
12		updjud.g	. . . . . . 7 ⊢ (𝜑 → 𝐺:𝐵⟶𝐶)
13		ffvelrn 6826	. . . . . . . 8 ⊢ ((𝐺:𝐵⟶𝐶 ∧ (2nd ‘𝑥) ∈ 𝐵) → (𝐺‘(2nd ‘𝑥)) ∈ 𝐶)
14	13	ex 416	. . . . . . 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 512	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 𝐵)) → ((1st ‘𝑥) ≠ ∅ → (𝐺‘(2nd ‘𝑥)) ∈ 𝐶))
17	9, 16	syl5bir 246	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 𝐵)) → (¬ (1st ‘𝑥) = ∅ → (𝐺‘(2nd ‘𝑥)) ∈ 𝐶))
18	17	imp 410	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴 ⊔ 𝐵)) ∧ ¬ (1st ‘𝑥) = ∅) → (𝐺‘(2nd ‘𝑥)) ∈ 𝐶)
19	8, 18	ifclda 4459	. 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 6855	1 ⊢ (𝜑 → 𝐻:(𝐴 ⊔ 𝐵)⟶𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∅c0 4243  ifcif 4425   ↦ cmpt 5110  ⟶wf 6320  ‘cfv 6324  1^st c1st 7669  2^nd c2nd 7670   ⊔ cdju 9311

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-1st 7671  df-2nd 7672  df-1o 8085  df-dju 9314

This theorem is referenced by:  updjudhcoinlf  9345  updjudhcoinrg  9346  updjud  9347