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

Theorem caseinl 7120
Description: Applying the "case" construction to a left injection. (Contributed by Jim Kingdon, 15-Mar-2023.)
Hypotheses
Ref Expression
caseinl.f (𝜑𝐹 Fn 𝐵)
caseinl.g (𝜑 → Fun 𝐺)
caseinl.a (𝜑𝐴𝐵)
Assertion
Ref Expression
caseinl (𝜑 → (case(𝐹, 𝐺)‘(inl‘𝐴)) = (𝐹𝐴))

Proof of Theorem caseinl
StepHypRef Expression
1 df-case 7113 . . . 4 case(𝐹, 𝐺) = ((𝐹inl) ∪ (𝐺inr))
21fveq1i 5535 . . 3 (case(𝐹, 𝐺)‘(inl‘𝐴)) = (((𝐹inl) ∪ (𝐺inr))‘(inl‘𝐴))
3 caseinl.f . . . . . . 7 (𝜑𝐹 Fn 𝐵)
4 fnfun 5332 . . . . . . 7 (𝐹 Fn 𝐵 → Fun 𝐹)
53, 4syl 14 . . . . . 6 (𝜑 → Fun 𝐹)
6 djulf1o 7087 . . . . . . . 8 inl:V–1-1-onto→({∅} × V)
7 f1ocnv 5493 . . . . . . . 8 (inl:V–1-1-onto→({∅} × V) → inl:({∅} × V)–1-1-onto→V)
86, 7ax-mp 5 . . . . . . 7 inl:({∅} × V)–1-1-onto→V
9 f1ofun 5482 . . . . . . 7 (inl:({∅} × V)–1-1-onto→V → Fun inl)
108, 9ax-mp 5 . . . . . 6 Fun inl
11 funco 5275 . . . . . 6 ((Fun 𝐹 ∧ Fun inl) → Fun (𝐹inl))
125, 10, 11sylancl 413 . . . . 5 (𝜑 → Fun (𝐹inl))
13 dmco 5155 . . . . . 6 dom (𝐹inl) = (inl “ dom 𝐹)
14 df-inl 7076 . . . . . . . . . . 11 inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
1514funmpt2 5274 . . . . . . . . . 10 Fun inl
16 funrel 5252 . . . . . . . . . 10 (Fun inl → Rel inl)
1715, 16ax-mp 5 . . . . . . . . 9 Rel inl
18 dfrel2 5097 . . . . . . . . 9 (Rel inl ↔ inl = inl)
1917, 18mpbi 145 . . . . . . . 8 inl = inl
2019a1i 9 . . . . . . 7 (𝜑inl = inl)
21 fndm 5334 . . . . . . . 8 (𝐹 Fn 𝐵 → dom 𝐹 = 𝐵)
223, 21syl 14 . . . . . . 7 (𝜑 → dom 𝐹 = 𝐵)
2320, 22imaeq12d 4989 . . . . . 6 (𝜑 → (inl “ dom 𝐹) = (inl “ 𝐵))
2413, 23eqtrid 2234 . . . . 5 (𝜑 → dom (𝐹inl) = (inl “ 𝐵))
25 df-fn 5238 . . . . 5 ((𝐹inl) Fn (inl “ 𝐵) ↔ (Fun (𝐹inl) ∧ dom (𝐹inl) = (inl “ 𝐵)))
2612, 24, 25sylanbrc 417 . . . 4 (𝜑 → (𝐹inl) Fn (inl “ 𝐵))
27 caseinl.g . . . . . 6 (𝜑 → Fun 𝐺)
28 djurf1o 7088 . . . . . . . 8 inr:V–1-1-onto→({1o} × V)
29 f1ocnv 5493 . . . . . . . 8 (inr:V–1-1-onto→({1o} × V) → inr:({1o} × V)–1-1-onto→V)
3028, 29ax-mp 5 . . . . . . 7 inr:({1o} × V)–1-1-onto→V
31 f1ofun 5482 . . . . . . 7 (inr:({1o} × V)–1-1-onto→V → Fun inr)
3230, 31ax-mp 5 . . . . . 6 Fun inr
33 funco 5275 . . . . . 6 ((Fun 𝐺 ∧ Fun inr) → Fun (𝐺inr))
3427, 32, 33sylancl 413 . . . . 5 (𝜑 → Fun (𝐺inr))
35 dmco 5155 . . . . . . 7 dom (𝐺inr) = (inr “ dom 𝐺)
36 imacnvcnv 5111 . . . . . . 7 (inr “ dom 𝐺) = (inr “ dom 𝐺)
3735, 36eqtri 2210 . . . . . 6 dom (𝐺inr) = (inr “ dom 𝐺)
3837a1i 9 . . . . 5 (𝜑 → dom (𝐺inr) = (inr “ dom 𝐺))
39 df-fn 5238 . . . . 5 ((𝐺inr) Fn (inr “ dom 𝐺) ↔ (Fun (𝐺inr) ∧ dom (𝐺inr) = (inr “ dom 𝐺)))
4034, 38, 39sylanbrc 417 . . . 4 (𝜑 → (𝐺inr) Fn (inr “ dom 𝐺))
41 djuin 7093 . . . . 5 ((inl “ 𝐵) ∩ (inr “ dom 𝐺)) = ∅
4241a1i 9 . . . 4 (𝜑 → ((inl “ 𝐵) ∩ (inr “ dom 𝐺)) = ∅)
43 caseinl.a . . . . . . . 8 (𝜑𝐴𝐵)
4443elexd 2765 . . . . . . 7 (𝜑𝐴 ∈ V)
45 f1odm 5484 . . . . . . . 8 (inl:V–1-1-onto→({∅} × V) → dom inl = V)
466, 45ax-mp 5 . . . . . . 7 dom inl = V
4744, 46eleqtrrdi 2283 . . . . . 6 (𝜑𝐴 ∈ dom inl)
4847, 15jctil 312 . . . . 5 (𝜑 → (Fun inl ∧ 𝐴 ∈ dom inl))
49 funfvima 5769 . . . . 5 ((Fun inl ∧ 𝐴 ∈ dom inl) → (𝐴𝐵 → (inl‘𝐴) ∈ (inl “ 𝐵)))
5048, 43, 49sylc 62 . . . 4 (𝜑 → (inl‘𝐴) ∈ (inl “ 𝐵))
51 fvun1 5603 . . . 4 (((𝐹inl) Fn (inl “ 𝐵) ∧ (𝐺inr) Fn (inr “ dom 𝐺) ∧ (((inl “ 𝐵) ∩ (inr “ dom 𝐺)) = ∅ ∧ (inl‘𝐴) ∈ (inl “ 𝐵))) → (((𝐹inl) ∪ (𝐺inr))‘(inl‘𝐴)) = ((𝐹inl)‘(inl‘𝐴)))
5226, 40, 42, 50, 51syl112anc 1253 . . 3 (𝜑 → (((𝐹inl) ∪ (𝐺inr))‘(inl‘𝐴)) = ((𝐹inl)‘(inl‘𝐴)))
532, 52eqtrid 2234 . 2 (𝜑 → (case(𝐹, 𝐺)‘(inl‘𝐴)) = ((𝐹inl)‘(inl‘𝐴)))
54 f1ofn 5481 . . . 4 (inl:({∅} × V)–1-1-onto→V → inl Fn ({∅} × V))
558, 54ax-mp 5 . . 3 inl Fn ({∅} × V)
56 f1of 5480 . . . . . 6 (inl:V–1-1-onto→({∅} × V) → inl:V⟶({∅} × V))
576, 56ax-mp 5 . . . . 5 inl:V⟶({∅} × V)
5857a1i 9 . . . 4 (𝜑 → inl:V⟶({∅} × V))
5958, 44ffvelcdmd 5673 . . 3 (𝜑 → (inl‘𝐴) ∈ ({∅} × V))
60 fvco2 5606 . . 3 ((inl Fn ({∅} × V) ∧ (inl‘𝐴) ∈ ({∅} × V)) → ((𝐹inl)‘(inl‘𝐴)) = (𝐹‘(inl‘(inl‘𝐴))))
6155, 59, 60sylancr 414 . 2 (𝜑 → ((𝐹inl)‘(inl‘𝐴)) = (𝐹‘(inl‘(inl‘𝐴))))
62 f1ocnvfv1 5799 . . . 4 ((inl:V–1-1-onto→({∅} × V) ∧ 𝐴 ∈ V) → (inl‘(inl‘𝐴)) = 𝐴)
636, 44, 62sylancr 414 . . 3 (𝜑 → (inl‘(inl‘𝐴)) = 𝐴)
6463fveq2d 5538 . 2 (𝜑 → (𝐹‘(inl‘(inl‘𝐴))) = (𝐹𝐴))
6553, 61, 643eqtrd 2226 1 (𝜑 → (case(𝐹, 𝐺)‘(inl‘𝐴)) = (𝐹𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wcel 2160  Vcvv 2752  cun 3142  cin 3143  c0 3437  {csn 3607  cop 3610   × cxp 4642  ccnv 4643  dom cdm 4644  cima 4647  ccom 4648  Rel wrel 4649  Fun wfun 5229   Fn wfn 5230  wf 5231  1-1-ontowf1o 5234  cfv 5235  1oc1o 6434  inlcinl 7074  inrcinr 7075  casecdjucase 7112
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-1st 6165  df-2nd 6166  df-1o 6441  df-inl 7076  df-inr 7077  df-case 7113
This theorem is referenced by:  omp1eomlem  7123  ctm  7138
  Copyright terms: Public domain W3C validator