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

Theorem updjudhf 6964
 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 6957 . . . . . 6 ((𝑥 ∈ (𝐴𝐵) ∧ (1st𝑥) = ∅) → (2nd𝑥) ∈ 𝐴)
21ex 114 . . . . 5 (𝑥 ∈ (𝐴𝐵) → ((1st𝑥) = ∅ → (2nd𝑥) ∈ 𝐴))
3 updjud.f . . . . . 6 (𝜑𝐹:𝐴𝐶)
4 ffvelrn 5553 . . . . . . 7 ((𝐹:𝐴𝐶 ∧ (2nd𝑥) ∈ 𝐴) → (𝐹‘(2nd𝑥)) ∈ 𝐶)
54ex 114 . . . . . 6 (𝐹:𝐴𝐶 → ((2nd𝑥) ∈ 𝐴 → (𝐹‘(2nd𝑥)) ∈ 𝐶))
63, 5syl 14 . . . . 5 (𝜑 → ((2nd𝑥) ∈ 𝐴 → (𝐹‘(2nd𝑥)) ∈ 𝐶))
72, 6sylan9r 407 . . . 4 ((𝜑𝑥 ∈ (𝐴𝐵)) → ((1st𝑥) = ∅ → (𝐹‘(2nd𝑥)) ∈ 𝐶))
87imp 123 . . 3 (((𝜑𝑥 ∈ (𝐴𝐵)) ∧ (1st𝑥) = ∅) → (𝐹‘(2nd𝑥)) ∈ 𝐶)
9 df-ne 2309 . . . . 5 ((1st𝑥) ≠ ∅ ↔ ¬ (1st𝑥) = ∅)
10 eldju2ndr 6958 . . . . . . 7 ((𝑥 ∈ (𝐴𝐵) ∧ (1st𝑥) ≠ ∅) → (2nd𝑥) ∈ 𝐵)
1110ex 114 . . . . . 6 (𝑥 ∈ (𝐴𝐵) → ((1st𝑥) ≠ ∅ → (2nd𝑥) ∈ 𝐵))
12 updjud.g . . . . . . 7 (𝜑𝐺:𝐵𝐶)
13 ffvelrn 5553 . . . . . . . 8 ((𝐺:𝐵𝐶 ∧ (2nd𝑥) ∈ 𝐵) → (𝐺‘(2nd𝑥)) ∈ 𝐶)
1413ex 114 . . . . . . 7 (𝐺:𝐵𝐶 → ((2nd𝑥) ∈ 𝐵 → (𝐺‘(2nd𝑥)) ∈ 𝐶))
1512, 14syl 14 . . . . . 6 (𝜑 → ((2nd𝑥) ∈ 𝐵 → (𝐺‘(2nd𝑥)) ∈ 𝐶))
1611, 15sylan9r 407 . . . . 5 ((𝜑𝑥 ∈ (𝐴𝐵)) → ((1st𝑥) ≠ ∅ → (𝐺‘(2nd𝑥)) ∈ 𝐶))
179, 16syl5bir 152 . . . 4 ((𝜑𝑥 ∈ (𝐴𝐵)) → (¬ (1st𝑥) = ∅ → (𝐺‘(2nd𝑥)) ∈ 𝐶))
1817imp 123 . . 3 (((𝜑𝑥 ∈ (𝐴𝐵)) ∧ ¬ (1st𝑥) = ∅) → (𝐺‘(2nd𝑥)) ∈ 𝐶)
19 eldju1st 6956 . . . . . 6 (𝑥 ∈ (𝐴𝐵) → ((1st𝑥) = ∅ ∨ (1st𝑥) = 1o))
20 1n0 6329 . . . . . . . 8 1o ≠ ∅
21 neeq1 2321 . . . . . . . 8 ((1st𝑥) = 1o → ((1st𝑥) ≠ ∅ ↔ 1o ≠ ∅))
2220, 21mpbiri 167 . . . . . . 7 ((1st𝑥) = 1o → (1st𝑥) ≠ ∅)
2322orim2i 750 . . . . . 6 (((1st𝑥) = ∅ ∨ (1st𝑥) = 1o) → ((1st𝑥) = ∅ ∨ (1st𝑥) ≠ ∅))
2419, 23syl 14 . . . . 5 (𝑥 ∈ (𝐴𝐵) → ((1st𝑥) = ∅ ∨ (1st𝑥) ≠ ∅))
2524adantl 275 . . . 4 ((𝜑𝑥 ∈ (𝐴𝐵)) → ((1st𝑥) = ∅ ∨ (1st𝑥) ≠ ∅))
26 dcne 2319 . . . 4 (DECID (1st𝑥) = ∅ ↔ ((1st𝑥) = ∅ ∨ (1st𝑥) ≠ ∅))
2725, 26sylibr 133 . . 3 ((𝜑𝑥 ∈ (𝐴𝐵)) → DECID (1st𝑥) = ∅)
288, 18, 27ifcldadc 3501 . 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 5574 1 (𝜑𝐻:(𝐴𝐵)⟶𝐶)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 697  DECID wdc 819   = wceq 1331   ∈ wcel 1480   ≠ wne 2308  ∅c0 3363  ifcif 3474   ↦ cmpt 3989  ⟶wf 5119  ‘cfv 5123  1st c1st 6036  2nd c2nd 6037  1oc1o 6306   ⊔ cdju 6922 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355 This theorem depends on definitions:  df-bi 116  df-dc 820  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-1st 6038  df-2nd 6039  df-1o 6313  df-dju 6923  df-inl 6932  df-inr 6933 This theorem is referenced by:  updjudhcoinlf  6965  updjudhcoinrg  6966  updjud  6967
 Copyright terms: Public domain W3C validator