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Theorem updjudhf 9971
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 9964 . . . . . 6 ((𝑥 ∈ (𝐴𝐵) ∧ (1st𝑥) = ∅) → (2nd𝑥) ∈ 𝐴)
21ex 412 . . . . 5 (𝑥 ∈ (𝐴𝐵) → ((1st𝑥) = ∅ → (2nd𝑥) ∈ 𝐴))
3 updjud.f . . . . . 6 (𝜑𝐹:𝐴𝐶)
4 ffvelcdm 7101 . . . . . . 7 ((𝐹:𝐴𝐶 ∧ (2nd𝑥) ∈ 𝐴) → (𝐹‘(2nd𝑥)) ∈ 𝐶)
54ex 412 . . . . . 6 (𝐹:𝐴𝐶 → ((2nd𝑥) ∈ 𝐴 → (𝐹‘(2nd𝑥)) ∈ 𝐶))
63, 5syl 17 . . . . 5 (𝜑 → ((2nd𝑥) ∈ 𝐴 → (𝐹‘(2nd𝑥)) ∈ 𝐶))
72, 6sylan9r 508 . . . 4 ((𝜑𝑥 ∈ (𝐴𝐵)) → ((1st𝑥) = ∅ → (𝐹‘(2nd𝑥)) ∈ 𝐶))
87imp 406 . . 3 (((𝜑𝑥 ∈ (𝐴𝐵)) ∧ (1st𝑥) = ∅) → (𝐹‘(2nd𝑥)) ∈ 𝐶)
9 df-ne 2941 . . . . 5 ((1st𝑥) ≠ ∅ ↔ ¬ (1st𝑥) = ∅)
10 eldju2ndr 9965 . . . . . . 7 ((𝑥 ∈ (𝐴𝐵) ∧ (1st𝑥) ≠ ∅) → (2nd𝑥) ∈ 𝐵)
1110ex 412 . . . . . 6 (𝑥 ∈ (𝐴𝐵) → ((1st𝑥) ≠ ∅ → (2nd𝑥) ∈ 𝐵))
12 updjud.g . . . . . . 7 (𝜑𝐺:𝐵𝐶)
13 ffvelcdm 7101 . . . . . . . 8 ((𝐺:𝐵𝐶 ∧ (2nd𝑥) ∈ 𝐵) → (𝐺‘(2nd𝑥)) ∈ 𝐶)
1413ex 412 . . . . . . 7 (𝐺:𝐵𝐶 → ((2nd𝑥) ∈ 𝐵 → (𝐺‘(2nd𝑥)) ∈ 𝐶))
1512, 14syl 17 . . . . . 6 (𝜑 → ((2nd𝑥) ∈ 𝐵 → (𝐺‘(2nd𝑥)) ∈ 𝐶))
1611, 15sylan9r 508 . . . . 5 ((𝜑𝑥 ∈ (𝐴𝐵)) → ((1st𝑥) ≠ ∅ → (𝐺‘(2nd𝑥)) ∈ 𝐶))
179, 16biimtrrid 243 . . . 4 ((𝜑𝑥 ∈ (𝐴𝐵)) → (¬ (1st𝑥) = ∅ → (𝐺‘(2nd𝑥)) ∈ 𝐶))
1817imp 406 . . 3 (((𝜑𝑥 ∈ (𝐴𝐵)) ∧ ¬ (1st𝑥) = ∅) → (𝐺‘(2nd𝑥)) ∈ 𝐶)
198, 18ifclda 4561 . 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 7134 1 (𝜑𝐻:(𝐴𝐵)⟶𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wcel 2108  wne 2940  c0 4333  ifcif 4525  cmpt 5225  wf 6557  cfv 6561  1st c1st 8012  2nd c2nd 8013  cdju 9938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-1st 8014  df-2nd 8015  df-1o 8506  df-dju 9941
This theorem is referenced by:  updjudhcoinlf  9972  updjudhcoinrg  9973  updjud  9974
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