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Theorem caseinr 6927
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 6919 . . . 4 case(𝐹, 𝐺) = ((𝐹inl) ∪ (𝐺inr))
21fveq1i 5374 . . 3 (case(𝐹, 𝐺)‘(inr‘𝐴)) = (((𝐹inl) ∪ (𝐺inr))‘(inr‘𝐴))
3 caseinr.f . . . . . 6 (𝜑 → Fun 𝐹)
4 djulf1o 6893 . . . . . . . 8 inl:V–1-1-onto→({∅} × V)
5 f1ocnv 5334 . . . . . . . 8 (inl:V–1-1-onto→({∅} × V) → inl:({∅} × V)–1-1-onto→V)
64, 5ax-mp 7 . . . . . . 7 inl:({∅} × V)–1-1-onto→V
7 f1ofun 5323 . . . . . . 7 (inl:({∅} × V)–1-1-onto→V → Fun inl)
86, 7ax-mp 7 . . . . . 6 Fun inl
9 funco 5119 . . . . . 6 ((Fun 𝐹 ∧ Fun inl) → Fun (𝐹inl))
103, 8, 9sylancl 407 . . . . 5 (𝜑 → Fun (𝐹inl))
11 dmco 5003 . . . . . . 7 dom (𝐹inl) = (inl “ dom 𝐹)
12 imacnvcnv 4959 . . . . . . 7 (inl “ dom 𝐹) = (inl “ dom 𝐹)
1311, 12eqtri 2133 . . . . . 6 dom (𝐹inl) = (inl “ dom 𝐹)
1413a1i 9 . . . . 5 (𝜑 → dom (𝐹inl) = (inl “ dom 𝐹))
15 df-fn 5082 . . . . 5 ((𝐹inl) Fn (inl “ dom 𝐹) ↔ (Fun (𝐹inl) ∧ dom (𝐹inl) = (inl “ dom 𝐹)))
1610, 14, 15sylanbrc 411 . . . 4 (𝜑 → (𝐹inl) Fn (inl “ dom 𝐹))
17 caseinr.g . . . . . . 7 (𝜑𝐺 Fn 𝐵)
18 fnfun 5176 . . . . . . 7 (𝐺 Fn 𝐵 → Fun 𝐺)
1917, 18syl 14 . . . . . 6 (𝜑 → Fun 𝐺)
20 djurf1o 6894 . . . . . . . 8 inr:V–1-1-onto→({1o} × V)
21 f1ocnv 5334 . . . . . . . 8 (inr:V–1-1-onto→({1o} × V) → inr:({1o} × V)–1-1-onto→V)
2220, 21ax-mp 7 . . . . . . 7 inr:({1o} × V)–1-1-onto→V
23 f1ofun 5323 . . . . . . 7 (inr:({1o} × V)–1-1-onto→V → Fun inr)
2422, 23ax-mp 7 . . . . . 6 Fun inr
25 funco 5119 . . . . . 6 ((Fun 𝐺 ∧ Fun inr) → Fun (𝐺inr))
2619, 24, 25sylancl 407 . . . . 5 (𝜑 → Fun (𝐺inr))
27 dmco 5003 . . . . . 6 dom (𝐺inr) = (inr “ dom 𝐺)
28 df-inr 6883 . . . . . . . . . . 11 inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
2928funmpt2 5118 . . . . . . . . . 10 Fun inr
30 funrel 5096 . . . . . . . . . 10 (Fun inr → Rel inr)
3129, 30ax-mp 7 . . . . . . . . 9 Rel inr
32 dfrel2 4945 . . . . . . . . 9 (Rel inr ↔ inr = inr)
3331, 32mpbi 144 . . . . . . . 8 inr = inr
3433a1i 9 . . . . . . 7 (𝜑inr = inr)
35 fndm 5178 . . . . . . . 8 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
3617, 35syl 14 . . . . . . 7 (𝜑 → dom 𝐺 = 𝐵)
3734, 36imaeq12d 4838 . . . . . 6 (𝜑 → (inr “ dom 𝐺) = (inr “ 𝐵))
3827, 37syl5eq 2157 . . . . 5 (𝜑 → dom (𝐺inr) = (inr “ 𝐵))
39 df-fn 5082 . . . . 5 ((𝐺inr) Fn (inr “ 𝐵) ↔ (Fun (𝐺inr) ∧ dom (𝐺inr) = (inr “ 𝐵)))
4026, 38, 39sylanbrc 411 . . . 4 (𝜑 → (𝐺inr) Fn (inr “ 𝐵))
41 djuin 6899 . . . . 5 ((inl “ dom 𝐹) ∩ (inr “ 𝐵)) = ∅
4241a1i 9 . . . 4 (𝜑 → ((inl “ dom 𝐹) ∩ (inr “ 𝐵)) = ∅)
43 caseinr.a . . . . . . . 8 (𝜑𝐴𝐵)
4443elexd 2668 . . . . . . 7 (𝜑𝐴 ∈ V)
45 f1odm 5325 . . . . . . . 8 (inr:V–1-1-onto→({1o} × V) → dom inr = V)
4620, 45ax-mp 7 . . . . . . 7 dom inr = V
4744, 46syl6eleqr 2206 . . . . . 6 (𝜑𝐴 ∈ dom inr)
4847, 29jctil 308 . . . . 5 (𝜑 → (Fun inr ∧ 𝐴 ∈ dom inr))
49 funfvima 5601 . . . . 5 ((Fun inr ∧ 𝐴 ∈ dom inr) → (𝐴𝐵 → (inr‘𝐴) ∈ (inr “ 𝐵)))
5048, 43, 49sylc 62 . . . 4 (𝜑 → (inr‘𝐴) ∈ (inr “ 𝐵))
51 fvun2 5440 . . . 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 1201 . . 3 (𝜑 → (((𝐹inl) ∪ (𝐺inr))‘(inr‘𝐴)) = ((𝐺inr)‘(inr‘𝐴)))
532, 52syl5eq 2157 . 2 (𝜑 → (case(𝐹, 𝐺)‘(inr‘𝐴)) = ((𝐺inr)‘(inr‘𝐴)))
54 f1ofn 5322 . . . 4 (inr:({1o} × V)–1-1-onto→V → inr Fn ({1o} × V))
5522, 54ax-mp 7 . . 3 inr Fn ({1o} × V)
56 f1of 5321 . . . . . 6 (inr:V–1-1-onto→({1o} × V) → inr:V⟶({1o} × V))
5720, 56ax-mp 7 . . . . 5 inr:V⟶({1o} × V)
5857a1i 9 . . . 4 (𝜑 → inr:V⟶({1o} × V))
5958, 44ffvelrnd 5508 . . 3 (𝜑 → (inr‘𝐴) ∈ ({1o} × V))
60 fvco2 5442 . . 3 ((inr Fn ({1o} × V) ∧ (inr‘𝐴) ∈ ({1o} × V)) → ((𝐺inr)‘(inr‘𝐴)) = (𝐺‘(inr‘(inr‘𝐴))))
6155, 59, 60sylancr 408 . 2 (𝜑 → ((𝐺inr)‘(inr‘𝐴)) = (𝐺‘(inr‘(inr‘𝐴))))
62 f1ocnvfv1 5630 . . . 4 ((inr:V–1-1-onto→({1o} × V) ∧ 𝐴 ∈ V) → (inr‘(inr‘𝐴)) = 𝐴)
6320, 44, 62sylancr 408 . . 3 (𝜑 → (inr‘(inr‘𝐴)) = 𝐴)
6463fveq2d 5377 . 2 (𝜑 → (𝐺‘(inr‘(inr‘𝐴))) = (𝐺𝐴))
6553, 61, 643eqtrd 2149 1 (𝜑 → (case(𝐹, 𝐺)‘(inr‘𝐴)) = (𝐺𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1312  wcel 1461  Vcvv 2655  cun 3033  cin 3034  c0 3327  {csn 3491  cop 3494   × cxp 4495  ccnv 4496  dom cdm 4497  cima 4500  ccom 4501  Rel wrel 4502  Fun wfun 5073   Fn wfn 5074  wf 5075  1-1-ontowf1o 5078  cfv 5079  1oc1o 6258  inlcinl 6880  inrcinr 6881  casecdjucase 6918
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-ral 2393  df-rex 2394  df-v 2657  df-sbc 2877  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-mpt 3949  df-tr 3985  df-id 4173  df-iord 4246  df-on 4248  df-suc 4251  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-1st 5990  df-2nd 5991  df-1o 6265  df-inl 6882  df-inr 6883  df-case 6919
This theorem is referenced by:  omp1eomlem  6929
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