MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  updjudhf Structured version   Visualization version   GIF version

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 (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 9337 . . . . . 6 ((𝑥 ∈ (𝐴𝐵) ∧ (1st𝑥) = ∅) → (2nd𝑥) ∈ 𝐴)
21ex 416 . . . . 5 (𝑥 ∈ (𝐴𝐵) → ((1st𝑥) = ∅ → (2nd𝑥) ∈ 𝐴))
3 updjud.f . . . . . 6 (𝜑𝐹:𝐴𝐶)
4 ffvelrn 6830 . . . . . . 7 ((𝐹:𝐴𝐶 ∧ (2nd𝑥) ∈ 𝐴) → (𝐹‘(2nd𝑥)) ∈ 𝐶)
54ex 416 . . . . . 6 (𝐹:𝐴𝐶 → ((2nd𝑥) ∈ 𝐴 → (𝐹‘(2nd𝑥)) ∈ 𝐶))
63, 5syl 17 . . . . 5 (𝜑 → ((2nd𝑥) ∈ 𝐴 → (𝐹‘(2nd𝑥)) ∈ 𝐶))
72, 6sylan9r 512 . . . 4 ((𝜑𝑥 ∈ (𝐴𝐵)) → ((1st𝑥) = ∅ → (𝐹‘(2nd𝑥)) ∈ 𝐶))
87imp 410 . . 3 (((𝜑𝑥 ∈ (𝐴𝐵)) ∧ (1st𝑥) = ∅) → (𝐹‘(2nd𝑥)) ∈ 𝐶)
9 df-ne 3014 . . . . 5 ((1st𝑥) ≠ ∅ ↔ ¬ (1st𝑥) = ∅)
10 eldju2ndr 9338 . . . . . . 7 ((𝑥 ∈ (𝐴𝐵) ∧ (1st𝑥) ≠ ∅) → (2nd𝑥) ∈ 𝐵)
1110ex 416 . . . . . 6 (𝑥 ∈ (𝐴𝐵) → ((1st𝑥) ≠ ∅ → (2nd𝑥) ∈ 𝐵))
12 updjud.g . . . . . . 7 (𝜑𝐺:𝐵𝐶)
13 ffvelrn 6830 . . . . . . . 8 ((𝐺:𝐵𝐶 ∧ (2nd𝑥) ∈ 𝐵) → (𝐺‘(2nd𝑥)) ∈ 𝐶)
1413ex 416 . . . . . . 7 (𝐺:𝐵𝐶 → ((2nd𝑥) ∈ 𝐵 → (𝐺‘(2nd𝑥)) ∈ 𝐶))
1512, 14syl 17 . . . . . 6 (𝜑 → ((2nd𝑥) ∈ 𝐵 → (𝐺‘(2nd𝑥)) ∈ 𝐶))
1611, 15sylan9r 512 . . . . 5 ((𝜑𝑥 ∈ (𝐴𝐵)) → ((1st𝑥) ≠ ∅ → (𝐺‘(2nd𝑥)) ∈ 𝐶))
179, 16syl5bir 246 . . . 4 ((𝜑𝑥 ∈ (𝐴𝐵)) → (¬ (1st𝑥) = ∅ → (𝐺‘(2nd𝑥)) ∈ 𝐶))
1817imp 410 . . 3 (((𝜑𝑥 ∈ (𝐴𝐵)) ∧ ¬ (1st𝑥) = ∅) → (𝐺‘(2nd𝑥)) ∈ 𝐶)
198, 18ifclda 4482 . 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 6859 1 (𝜑𝐻:(𝐴𝐵)⟶𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2115  wne 3013  c0 4274  ifcif 4448  cmpt 5127  wf 6332  cfv 6336  1st c1st 7670  2nd c2nd 7671  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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-suc 6178  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-fv 6344  df-1st 7672  df-2nd 7673  df-1o 8085  df-dju 9314
This theorem is referenced by:  updjudhcoinlf  9345  updjudhcoinrg  9346  updjud  9347
  Copyright terms: Public domain W3C validator