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Theorem caseinr 7122
Description: Applying the "case" construction to a right injection. (Contributed by Jim Kingdon, 12-Jul-2023.)
Hypotheses
Ref Expression
caseinr.f (𝜑 → Fun 𝐹)
caseinr.g (𝜑𝐺 Fn 𝐵)
caseinr.a (𝜑𝐴𝐵)
Assertion
Ref Expression
caseinr (𝜑 → (case(𝐹, 𝐺)‘(inr‘𝐴)) = (𝐺𝐴))

Proof of Theorem caseinr
StepHypRef Expression
1 df-case 7114 . . . 4 case(𝐹, 𝐺) = ((𝐹inl) ∪ (𝐺inr))
21fveq1i 5535 . . 3 (case(𝐹, 𝐺)‘(inr‘𝐴)) = (((𝐹inl) ∪ (𝐺inr))‘(inr‘𝐴))
3 caseinr.f . . . . . 6 (𝜑 → Fun 𝐹)
4 djulf1o 7088 . . . . . . . 8 inl:V–1-1-onto→({∅} × V)
5 f1ocnv 5493 . . . . . . . 8 (inl:V–1-1-onto→({∅} × V) → inl:({∅} × V)–1-1-onto→V)
64, 5ax-mp 5 . . . . . . 7 inl:({∅} × V)–1-1-onto→V
7 f1ofun 5482 . . . . . . 7 (inl:({∅} × V)–1-1-onto→V → Fun inl)
86, 7ax-mp 5 . . . . . 6 Fun inl
9 funco 5275 . . . . . 6 ((Fun 𝐹 ∧ Fun inl) → Fun (𝐹inl))
103, 8, 9sylancl 413 . . . . 5 (𝜑 → Fun (𝐹inl))
11 dmco 5155 . . . . . . 7 dom (𝐹inl) = (inl “ dom 𝐹)
12 imacnvcnv 5111 . . . . . . 7 (inl “ dom 𝐹) = (inl “ dom 𝐹)
1311, 12eqtri 2210 . . . . . 6 dom (𝐹inl) = (inl “ dom 𝐹)
1413a1i 9 . . . . 5 (𝜑 → dom (𝐹inl) = (inl “ dom 𝐹))
15 df-fn 5238 . . . . 5 ((𝐹inl) Fn (inl “ dom 𝐹) ↔ (Fun (𝐹inl) ∧ dom (𝐹inl) = (inl “ dom 𝐹)))
1610, 14, 15sylanbrc 417 . . . 4 (𝜑 → (𝐹inl) Fn (inl “ dom 𝐹))
17 caseinr.g . . . . . . 7 (𝜑𝐺 Fn 𝐵)
18 fnfun 5332 . . . . . . 7 (𝐺 Fn 𝐵 → Fun 𝐺)
1917, 18syl 14 . . . . . 6 (𝜑 → Fun 𝐺)
20 djurf1o 7089 . . . . . . . 8 inr:V–1-1-onto→({1o} × V)
21 f1ocnv 5493 . . . . . . . 8 (inr:V–1-1-onto→({1o} × V) → inr:({1o} × V)–1-1-onto→V)
2220, 21ax-mp 5 . . . . . . 7 inr:({1o} × V)–1-1-onto→V
23 f1ofun 5482 . . . . . . 7 (inr:({1o} × V)–1-1-onto→V → Fun inr)
2422, 23ax-mp 5 . . . . . 6 Fun inr
25 funco 5275 . . . . . 6 ((Fun 𝐺 ∧ Fun inr) → Fun (𝐺inr))
2619, 24, 25sylancl 413 . . . . 5 (𝜑 → Fun (𝐺inr))
27 dmco 5155 . . . . . 6 dom (𝐺inr) = (inr “ dom 𝐺)
28 df-inr 7078 . . . . . . . . . . 11 inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
2928funmpt2 5274 . . . . . . . . . 10 Fun inr
30 funrel 5252 . . . . . . . . . 10 (Fun inr → Rel inr)
3129, 30ax-mp 5 . . . . . . . . 9 Rel inr
32 dfrel2 5097 . . . . . . . . 9 (Rel inr ↔ inr = inr)
3331, 32mpbi 145 . . . . . . . 8 inr = inr
3433a1i 9 . . . . . . 7 (𝜑inr = inr)
35 fndm 5334 . . . . . . . 8 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
3617, 35syl 14 . . . . . . 7 (𝜑 → dom 𝐺 = 𝐵)
3734, 36imaeq12d 4989 . . . . . 6 (𝜑 → (inr “ dom 𝐺) = (inr “ 𝐵))
3827, 37eqtrid 2234 . . . . 5 (𝜑 → dom (𝐺inr) = (inr “ 𝐵))
39 df-fn 5238 . . . . 5 ((𝐺inr) Fn (inr “ 𝐵) ↔ (Fun (𝐺inr) ∧ dom (𝐺inr) = (inr “ 𝐵)))
4026, 38, 39sylanbrc 417 . . . 4 (𝜑 → (𝐺inr) Fn (inr “ 𝐵))
41 djuin 7094 . . . . 5 ((inl “ dom 𝐹) ∩ (inr “ 𝐵)) = ∅
4241a1i 9 . . . 4 (𝜑 → ((inl “ dom 𝐹) ∩ (inr “ 𝐵)) = ∅)
43 caseinr.a . . . . . . . 8 (𝜑𝐴𝐵)
4443elexd 2765 . . . . . . 7 (𝜑𝐴 ∈ V)
45 f1odm 5484 . . . . . . . 8 (inr:V–1-1-onto→({1o} × V) → dom inr = V)
4620, 45ax-mp 5 . . . . . . 7 dom inr = V
4744, 46eleqtrrdi 2283 . . . . . 6 (𝜑𝐴 ∈ dom inr)
4847, 29jctil 312 . . . . 5 (𝜑 → (Fun inr ∧ 𝐴 ∈ dom inr))
49 funfvima 5769 . . . . 5 ((Fun inr ∧ 𝐴 ∈ dom inr) → (𝐴𝐵 → (inr‘𝐴) ∈ (inr “ 𝐵)))
5048, 43, 49sylc 62 . . . 4 (𝜑 → (inr‘𝐴) ∈ (inr “ 𝐵))
51 fvun2 5604 . . . 4 (((𝐹inl) Fn (inl “ dom 𝐹) ∧ (𝐺inr) Fn (inr “ 𝐵) ∧ (((inl “ dom 𝐹) ∩ (inr “ 𝐵)) = ∅ ∧ (inr‘𝐴) ∈ (inr “ 𝐵))) → (((𝐹inl) ∪ (𝐺inr))‘(inr‘𝐴)) = ((𝐺inr)‘(inr‘𝐴)))
5216, 40, 42, 50, 51syl112anc 1253 . . 3 (𝜑 → (((𝐹inl) ∪ (𝐺inr))‘(inr‘𝐴)) = ((𝐺inr)‘(inr‘𝐴)))
532, 52eqtrid 2234 . 2 (𝜑 → (case(𝐹, 𝐺)‘(inr‘𝐴)) = ((𝐺inr)‘(inr‘𝐴)))
54 f1ofn 5481 . . . 4 (inr:({1o} × V)–1-1-onto→V → inr Fn ({1o} × V))
5522, 54ax-mp 5 . . 3 inr Fn ({1o} × V)
56 f1of 5480 . . . . . 6 (inr:V–1-1-onto→({1o} × V) → inr:V⟶({1o} × V))
5720, 56ax-mp 5 . . . . 5 inr:V⟶({1o} × V)
5857a1i 9 . . . 4 (𝜑 → inr:V⟶({1o} × V))
5958, 44ffvelcdmd 5673 . . 3 (𝜑 → (inr‘𝐴) ∈ ({1o} × V))
60 fvco2 5606 . . 3 ((inr Fn ({1o} × V) ∧ (inr‘𝐴) ∈ ({1o} × V)) → ((𝐺inr)‘(inr‘𝐴)) = (𝐺‘(inr‘(inr‘𝐴))))
6155, 59, 60sylancr 414 . 2 (𝜑 → ((𝐺inr)‘(inr‘𝐴)) = (𝐺‘(inr‘(inr‘𝐴))))
62 f1ocnvfv1 5799 . . . 4 ((inr:V–1-1-onto→({1o} × V) ∧ 𝐴 ∈ V) → (inr‘(inr‘𝐴)) = 𝐴)
6320, 44, 62sylancr 414 . . 3 (𝜑 → (inr‘(inr‘𝐴)) = 𝐴)
6463fveq2d 5538 . 2 (𝜑 → (𝐺‘(inr‘(inr‘𝐴))) = (𝐺𝐴))
6553, 61, 643eqtrd 2226 1 (𝜑 → (case(𝐹, 𝐺)‘(inr‘𝐴)) = (𝐺𝐴))
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 6435  inlcinl 7075  inrcinr 7076  casecdjucase 7113
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 6166  df-2nd 6167  df-1o 6442  df-inl 7077  df-inr 7078  df-case 7114
This theorem is referenced by:  omp1eomlem  7124
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