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Theorem caseinr 6985
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 6977 . . . 4 case(𝐹, 𝐺) = ((𝐹inl) ∪ (𝐺inr))
21fveq1i 5430 . . 3 (case(𝐹, 𝐺)‘(inr‘𝐴)) = (((𝐹inl) ∪ (𝐺inr))‘(inr‘𝐴))
3 caseinr.f . . . . . 6 (𝜑 → Fun 𝐹)
4 djulf1o 6951 . . . . . . . 8 inl:V–1-1-onto→({∅} × V)
5 f1ocnv 5388 . . . . . . . 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 5377 . . . . . . 7 (inl:({∅} × V)–1-1-onto→V → Fun inl)
86, 7ax-mp 5 . . . . . 6 Fun inl
9 funco 5171 . . . . . 6 ((Fun 𝐹 ∧ Fun inl) → Fun (𝐹inl))
103, 8, 9sylancl 410 . . . . 5 (𝜑 → Fun (𝐹inl))
11 dmco 5055 . . . . . . 7 dom (𝐹inl) = (inl “ dom 𝐹)
12 imacnvcnv 5011 . . . . . . 7 (inl “ dom 𝐹) = (inl “ dom 𝐹)
1311, 12eqtri 2161 . . . . . 6 dom (𝐹inl) = (inl “ dom 𝐹)
1413a1i 9 . . . . 5 (𝜑 → dom (𝐹inl) = (inl “ dom 𝐹))
15 df-fn 5134 . . . . 5 ((𝐹inl) Fn (inl “ dom 𝐹) ↔ (Fun (𝐹inl) ∧ dom (𝐹inl) = (inl “ dom 𝐹)))
1610, 14, 15sylanbrc 414 . . . 4 (𝜑 → (𝐹inl) Fn (inl “ dom 𝐹))
17 caseinr.g . . . . . . 7 (𝜑𝐺 Fn 𝐵)
18 fnfun 5228 . . . . . . 7 (𝐺 Fn 𝐵 → Fun 𝐺)
1917, 18syl 14 . . . . . 6 (𝜑 → Fun 𝐺)
20 djurf1o 6952 . . . . . . . 8 inr:V–1-1-onto→({1o} × V)
21 f1ocnv 5388 . . . . . . . 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 5377 . . . . . . 7 (inr:({1o} × V)–1-1-onto→V → Fun inr)
2422, 23ax-mp 5 . . . . . 6 Fun inr
25 funco 5171 . . . . . 6 ((Fun 𝐺 ∧ Fun inr) → Fun (𝐺inr))
2619, 24, 25sylancl 410 . . . . 5 (𝜑 → Fun (𝐺inr))
27 dmco 5055 . . . . . 6 dom (𝐺inr) = (inr “ dom 𝐺)
28 df-inr 6941 . . . . . . . . . . 11 inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
2928funmpt2 5170 . . . . . . . . . 10 Fun inr
30 funrel 5148 . . . . . . . . . 10 (Fun inr → Rel inr)
3129, 30ax-mp 5 . . . . . . . . 9 Rel inr
32 dfrel2 4997 . . . . . . . . 9 (Rel inr ↔ inr = inr)
3331, 32mpbi 144 . . . . . . . 8 inr = inr
3433a1i 9 . . . . . . 7 (𝜑inr = inr)
35 fndm 5230 . . . . . . . 8 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
3617, 35syl 14 . . . . . . 7 (𝜑 → dom 𝐺 = 𝐵)
3734, 36imaeq12d 4890 . . . . . 6 (𝜑 → (inr “ dom 𝐺) = (inr “ 𝐵))
3827, 37syl5eq 2185 . . . . 5 (𝜑 → dom (𝐺inr) = (inr “ 𝐵))
39 df-fn 5134 . . . . 5 ((𝐺inr) Fn (inr “ 𝐵) ↔ (Fun (𝐺inr) ∧ dom (𝐺inr) = (inr “ 𝐵)))
4026, 38, 39sylanbrc 414 . . . 4 (𝜑 → (𝐺inr) Fn (inr “ 𝐵))
41 djuin 6957 . . . . 5 ((inl “ dom 𝐹) ∩ (inr “ 𝐵)) = ∅
4241a1i 9 . . . 4 (𝜑 → ((inl “ dom 𝐹) ∩ (inr “ 𝐵)) = ∅)
43 caseinr.a . . . . . . . 8 (𝜑𝐴𝐵)
4443elexd 2702 . . . . . . 7 (𝜑𝐴 ∈ V)
45 f1odm 5379 . . . . . . . 8 (inr:V–1-1-onto→({1o} × V) → dom inr = V)
4620, 45ax-mp 5 . . . . . . 7 dom inr = V
4744, 46eleqtrrdi 2234 . . . . . 6 (𝜑𝐴 ∈ dom inr)
4847, 29jctil 310 . . . . 5 (𝜑 → (Fun inr ∧ 𝐴 ∈ dom inr))
49 funfvima 5657 . . . . 5 ((Fun inr ∧ 𝐴 ∈ dom inr) → (𝐴𝐵 → (inr‘𝐴) ∈ (inr “ 𝐵)))
5048, 43, 49sylc 62 . . . 4 (𝜑 → (inr‘𝐴) ∈ (inr “ 𝐵))
51 fvun2 5496 . . . 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 1221 . . 3 (𝜑 → (((𝐹inl) ∪ (𝐺inr))‘(inr‘𝐴)) = ((𝐺inr)‘(inr‘𝐴)))
532, 52syl5eq 2185 . 2 (𝜑 → (case(𝐹, 𝐺)‘(inr‘𝐴)) = ((𝐺inr)‘(inr‘𝐴)))
54 f1ofn 5376 . . . 4 (inr:({1o} × V)–1-1-onto→V → inr Fn ({1o} × V))
5522, 54ax-mp 5 . . 3 inr Fn ({1o} × V)
56 f1of 5375 . . . . . 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, 44ffvelrnd 5564 . . 3 (𝜑 → (inr‘𝐴) ∈ ({1o} × V))
60 fvco2 5498 . . 3 ((inr Fn ({1o} × V) ∧ (inr‘𝐴) ∈ ({1o} × V)) → ((𝐺inr)‘(inr‘𝐴)) = (𝐺‘(inr‘(inr‘𝐴))))
6155, 59, 60sylancr 411 . 2 (𝜑 → ((𝐺inr)‘(inr‘𝐴)) = (𝐺‘(inr‘(inr‘𝐴))))
62 f1ocnvfv1 5686 . . . 4 ((inr:V–1-1-onto→({1o} × V) ∧ 𝐴 ∈ V) → (inr‘(inr‘𝐴)) = 𝐴)
6320, 44, 62sylancr 411 . . 3 (𝜑 → (inr‘(inr‘𝐴)) = 𝐴)
6463fveq2d 5433 . 2 (𝜑 → (𝐺‘(inr‘(inr‘𝐴))) = (𝐺𝐴))
6553, 61, 643eqtrd 2177 1 (𝜑 → (case(𝐹, 𝐺)‘(inr‘𝐴)) = (𝐺𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1332  wcel 1481  Vcvv 2689  cun 3074  cin 3075  c0 3368  {csn 3532  cop 3535   × cxp 4545  ccnv 4546  dom cdm 4547  cima 4550  ccom 4551  Rel wrel 4552  Fun wfun 5125   Fn wfn 5126  wf 5127  1-1-ontowf1o 5130  cfv 5131  1oc1o 6314  inlcinl 6938  inrcinr 6939  casecdjucase 6976
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-suc 4301  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-1st 6046  df-2nd 6047  df-1o 6321  df-inl 6940  df-inr 6941  df-case 6977
This theorem is referenced by:  omp1eomlem  6987
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