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Theorem updjudhf 9092
 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 (𝜑𝐹:𝐴𝐶)
updjud.g (𝜑𝐺:𝐵𝐶)
updjudhf.h 𝐻 = (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))
Assertion
Ref Expression
updjudhf (𝜑𝐻:(𝐴𝐵)⟶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
StepHypRef Expression
1 eldju2ndl 9085 . . . . . 6 ((𝑥 ∈ (𝐴𝐵) ∧ (1st𝑥) = ∅) → (2nd𝑥) ∈ 𝐴)
21ex 403 . . . . 5 (𝑥 ∈ (𝐴𝐵) → ((1st𝑥) = ∅ → (2nd𝑥) ∈ 𝐴))
3 updjud.f . . . . . 6 (𝜑𝐹:𝐴𝐶)
4 ffvelrn 6623 . . . . . . 7 ((𝐹:𝐴𝐶 ∧ (2nd𝑥) ∈ 𝐴) → (𝐹‘(2nd𝑥)) ∈ 𝐶)
54ex 403 . . . . . 6 (𝐹:𝐴𝐶 → ((2nd𝑥) ∈ 𝐴 → (𝐹‘(2nd𝑥)) ∈ 𝐶))
63, 5syl 17 . . . . 5 (𝜑 → ((2nd𝑥) ∈ 𝐴 → (𝐹‘(2nd𝑥)) ∈ 𝐶))
72, 6sylan9r 504 . . . 4 ((𝜑𝑥 ∈ (𝐴𝐵)) → ((1st𝑥) = ∅ → (𝐹‘(2nd𝑥)) ∈ 𝐶))
87imp 397 . . 3 (((𝜑𝑥 ∈ (𝐴𝐵)) ∧ (1st𝑥) = ∅) → (𝐹‘(2nd𝑥)) ∈ 𝐶)
9 df-ne 2970 . . . . 5 ((1st𝑥) ≠ ∅ ↔ ¬ (1st𝑥) = ∅)
10 eldju2ndr 9086 . . . . . . 7 ((𝑥 ∈ (𝐴𝐵) ∧ (1st𝑥) ≠ ∅) → (2nd𝑥) ∈ 𝐵)
1110ex 403 . . . . . 6 (𝑥 ∈ (𝐴𝐵) → ((1st𝑥) ≠ ∅ → (2nd𝑥) ∈ 𝐵))
12 updjud.g . . . . . . 7 (𝜑𝐺:𝐵𝐶)
13 ffvelrn 6623 . . . . . . . 8 ((𝐺:𝐵𝐶 ∧ (2nd𝑥) ∈ 𝐵) → (𝐺‘(2nd𝑥)) ∈ 𝐶)
1413ex 403 . . . . . . 7 (𝐺:𝐵𝐶 → ((2nd𝑥) ∈ 𝐵 → (𝐺‘(2nd𝑥)) ∈ 𝐶))
1512, 14syl 17 . . . . . 6 (𝜑 → ((2nd𝑥) ∈ 𝐵 → (𝐺‘(2nd𝑥)) ∈ 𝐶))
1611, 15sylan9r 504 . . . . 5 ((𝜑𝑥 ∈ (𝐴𝐵)) → ((1st𝑥) ≠ ∅ → (𝐺‘(2nd𝑥)) ∈ 𝐶))
179, 16syl5bir 235 . . . 4 ((𝜑𝑥 ∈ (𝐴𝐵)) → (¬ (1st𝑥) = ∅ → (𝐺‘(2nd𝑥)) ∈ 𝐶))
1817imp 397 . . 3 (((𝜑𝑥 ∈ (𝐴𝐵)) ∧ ¬ (1st𝑥) = ∅) → (𝐺‘(2nd𝑥)) ∈ 𝐶)
198, 18ifclda 4341 . 2 ((𝜑𝑥 ∈ (𝐴𝐵)) → if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))) ∈ 𝐶)
20 updjudhf.h . 2 𝐻 = (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))
2119, 20fmptd 6650 1 (𝜑𝐻:(𝐴𝐵)⟶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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