ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  updjudhf GIF version

Theorem updjudhf 7078
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 7071 . . . . . 6 ((𝑥 ∈ (𝐴𝐵) ∧ (1st𝑥) = ∅) → (2nd𝑥) ∈ 𝐴)
21ex 115 . . . . 5 (𝑥 ∈ (𝐴𝐵) → ((1st𝑥) = ∅ → (2nd𝑥) ∈ 𝐴))
3 updjud.f . . . . . 6 (𝜑𝐹:𝐴𝐶)
4 ffvelcdm 5650 . . . . . . 7 ((𝐹:𝐴𝐶 ∧ (2nd𝑥) ∈ 𝐴) → (𝐹‘(2nd𝑥)) ∈ 𝐶)
54ex 115 . . . . . 6 (𝐹:𝐴𝐶 → ((2nd𝑥) ∈ 𝐴 → (𝐹‘(2nd𝑥)) ∈ 𝐶))
63, 5syl 14 . . . . 5 (𝜑 → ((2nd𝑥) ∈ 𝐴 → (𝐹‘(2nd𝑥)) ∈ 𝐶))
72, 6sylan9r 410 . . . 4 ((𝜑𝑥 ∈ (𝐴𝐵)) → ((1st𝑥) = ∅ → (𝐹‘(2nd𝑥)) ∈ 𝐶))
87imp 124 . . 3 (((𝜑𝑥 ∈ (𝐴𝐵)) ∧ (1st𝑥) = ∅) → (𝐹‘(2nd𝑥)) ∈ 𝐶)
9 df-ne 2348 . . . . 5 ((1st𝑥) ≠ ∅ ↔ ¬ (1st𝑥) = ∅)
10 eldju2ndr 7072 . . . . . . 7 ((𝑥 ∈ (𝐴𝐵) ∧ (1st𝑥) ≠ ∅) → (2nd𝑥) ∈ 𝐵)
1110ex 115 . . . . . 6 (𝑥 ∈ (𝐴𝐵) → ((1st𝑥) ≠ ∅ → (2nd𝑥) ∈ 𝐵))
12 updjud.g . . . . . . 7 (𝜑𝐺:𝐵𝐶)
13 ffvelcdm 5650 . . . . . . . 8 ((𝐺:𝐵𝐶 ∧ (2nd𝑥) ∈ 𝐵) → (𝐺‘(2nd𝑥)) ∈ 𝐶)
1413ex 115 . . . . . . 7 (𝐺:𝐵𝐶 → ((2nd𝑥) ∈ 𝐵 → (𝐺‘(2nd𝑥)) ∈ 𝐶))
1512, 14syl 14 . . . . . 6 (𝜑 → ((2nd𝑥) ∈ 𝐵 → (𝐺‘(2nd𝑥)) ∈ 𝐶))
1611, 15sylan9r 410 . . . . 5 ((𝜑𝑥 ∈ (𝐴𝐵)) → ((1st𝑥) ≠ ∅ → (𝐺‘(2nd𝑥)) ∈ 𝐶))
179, 16biimtrrid 153 . . . 4 ((𝜑𝑥 ∈ (𝐴𝐵)) → (¬ (1st𝑥) = ∅ → (𝐺‘(2nd𝑥)) ∈ 𝐶))
1817imp 124 . . 3 (((𝜑𝑥 ∈ (𝐴𝐵)) ∧ ¬ (1st𝑥) = ∅) → (𝐺‘(2nd𝑥)) ∈ 𝐶)
19 eldju1st 7070 . . . . . 6 (𝑥 ∈ (𝐴𝐵) → ((1st𝑥) = ∅ ∨ (1st𝑥) = 1o))
20 1n0 6433 . . . . . . . 8 1o ≠ ∅
21 neeq1 2360 . . . . . . . 8 ((1st𝑥) = 1o → ((1st𝑥) ≠ ∅ ↔ 1o ≠ ∅))
2220, 21mpbiri 168 . . . . . . 7 ((1st𝑥) = 1o → (1st𝑥) ≠ ∅)
2322orim2i 761 . . . . . 6 (((1st𝑥) = ∅ ∨ (1st𝑥) = 1o) → ((1st𝑥) = ∅ ∨ (1st𝑥) ≠ ∅))
2419, 23syl 14 . . . . 5 (𝑥 ∈ (𝐴𝐵) → ((1st𝑥) = ∅ ∨ (1st𝑥) ≠ ∅))
2524adantl 277 . . . 4 ((𝜑𝑥 ∈ (𝐴𝐵)) → ((1st𝑥) = ∅ ∨ (1st𝑥) ≠ ∅))
26 dcne 2358 . . . 4 (DECID (1st𝑥) = ∅ ↔ ((1st𝑥) = ∅ ∨ (1st𝑥) ≠ ∅))
2725, 26sylibr 134 . . 3 ((𝜑𝑥 ∈ (𝐴𝐵)) → DECID (1st𝑥) = ∅)
288, 18, 27ifcldadc 3564 . 2 ((𝜑𝑥 ∈ (𝐴𝐵)) → if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))) ∈ 𝐶)
29 updjudhf.h . 2 𝐻 = (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))
3028, 29fmptd 5671 1 (𝜑𝐻:(𝐴𝐵)⟶𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wo 708  DECID wdc 834   = wceq 1353  wcel 2148  wne 2347  c0 3423  ifcif 3535  cmpt 4065  wf 5213  cfv 5217  1st c1st 6139  2nd c2nd 6140  1oc1o 6410  cdju 7036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-1st 6141  df-2nd 6142  df-1o 6417  df-dju 7037  df-inl 7046  df-inr 7047
This theorem is referenced by:  updjudhcoinlf  7079  updjudhcoinrg  7080  updjud  7081
  Copyright terms: Public domain W3C validator