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Theorem caseinr 7085
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 7077 . . . 4 case(𝐹, 𝐺) = ((𝐹inl) ∪ (𝐺inr))
21fveq1i 5512 . . 3 (case(𝐹, 𝐺)‘(inr‘𝐴)) = (((𝐹inl) ∪ (𝐺inr))‘(inr‘𝐴))
3 caseinr.f . . . . . 6 (𝜑 → Fun 𝐹)
4 djulf1o 7051 . . . . . . . 8 inl:V–1-1-onto→({∅} × V)
5 f1ocnv 5470 . . . . . . . 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 5459 . . . . . . 7 (inl:({∅} × V)–1-1-onto→V → Fun inl)
86, 7ax-mp 5 . . . . . 6 Fun inl
9 funco 5252 . . . . . 6 ((Fun 𝐹 ∧ Fun inl) → Fun (𝐹inl))
103, 8, 9sylancl 413 . . . . 5 (𝜑 → Fun (𝐹inl))
11 dmco 5133 . . . . . . 7 dom (𝐹inl) = (inl “ dom 𝐹)
12 imacnvcnv 5089 . . . . . . 7 (inl “ dom 𝐹) = (inl “ dom 𝐹)
1311, 12eqtri 2198 . . . . . 6 dom (𝐹inl) = (inl “ dom 𝐹)
1413a1i 9 . . . . 5 (𝜑 → dom (𝐹inl) = (inl “ dom 𝐹))
15 df-fn 5215 . . . . 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 5309 . . . . . . 7 (𝐺 Fn 𝐵 → Fun 𝐺)
1917, 18syl 14 . . . . . 6 (𝜑 → Fun 𝐺)
20 djurf1o 7052 . . . . . . . 8 inr:V–1-1-onto→({1o} × V)
21 f1ocnv 5470 . . . . . . . 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 5459 . . . . . . 7 (inr:({1o} × V)–1-1-onto→V → Fun inr)
2422, 23ax-mp 5 . . . . . 6 Fun inr
25 funco 5252 . . . . . 6 ((Fun 𝐺 ∧ Fun inr) → Fun (𝐺inr))
2619, 24, 25sylancl 413 . . . . 5 (𝜑 → Fun (𝐺inr))
27 dmco 5133 . . . . . 6 dom (𝐺inr) = (inr “ dom 𝐺)
28 df-inr 7041 . . . . . . . . . . 11 inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
2928funmpt2 5251 . . . . . . . . . 10 Fun inr
30 funrel 5229 . . . . . . . . . 10 (Fun inr → Rel inr)
3129, 30ax-mp 5 . . . . . . . . 9 Rel inr
32 dfrel2 5075 . . . . . . . . 9 (Rel inr ↔ inr = inr)
3331, 32mpbi 145 . . . . . . . 8 inr = inr
3433a1i 9 . . . . . . 7 (𝜑inr = inr)
35 fndm 5311 . . . . . . . 8 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
3617, 35syl 14 . . . . . . 7 (𝜑 → dom 𝐺 = 𝐵)
3734, 36imaeq12d 4967 . . . . . 6 (𝜑 → (inr “ dom 𝐺) = (inr “ 𝐵))
3827, 37eqtrid 2222 . . . . 5 (𝜑 → dom (𝐺inr) = (inr “ 𝐵))
39 df-fn 5215 . . . . 5 ((𝐺inr) Fn (inr “ 𝐵) ↔ (Fun (𝐺inr) ∧ dom (𝐺inr) = (inr “ 𝐵)))
4026, 38, 39sylanbrc 417 . . . 4 (𝜑 → (𝐺inr) Fn (inr “ 𝐵))
41 djuin 7057 . . . . 5 ((inl “ dom 𝐹) ∩ (inr “ 𝐵)) = ∅
4241a1i 9 . . . 4 (𝜑 → ((inl “ dom 𝐹) ∩ (inr “ 𝐵)) = ∅)
43 caseinr.a . . . . . . . 8 (𝜑𝐴𝐵)
4443elexd 2750 . . . . . . 7 (𝜑𝐴 ∈ V)
45 f1odm 5461 . . . . . . . 8 (inr:V–1-1-onto→({1o} × V) → dom inr = V)
4620, 45ax-mp 5 . . . . . . 7 dom inr = V
4744, 46eleqtrrdi 2271 . . . . . 6 (𝜑𝐴 ∈ dom inr)
4847, 29jctil 312 . . . . 5 (𝜑 → (Fun inr ∧ 𝐴 ∈ dom inr))
49 funfvima 5743 . . . . 5 ((Fun inr ∧ 𝐴 ∈ dom inr) → (𝐴𝐵 → (inr‘𝐴) ∈ (inr “ 𝐵)))
5048, 43, 49sylc 62 . . . 4 (𝜑 → (inr‘𝐴) ∈ (inr “ 𝐵))
51 fvun2 5579 . . . 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 1242 . . 3 (𝜑 → (((𝐹inl) ∪ (𝐺inr))‘(inr‘𝐴)) = ((𝐺inr)‘(inr‘𝐴)))
532, 52eqtrid 2222 . 2 (𝜑 → (case(𝐹, 𝐺)‘(inr‘𝐴)) = ((𝐺inr)‘(inr‘𝐴)))
54 f1ofn 5458 . . . 4 (inr:({1o} × V)–1-1-onto→V → inr Fn ({1o} × V))
5522, 54ax-mp 5 . . 3 inr Fn ({1o} × V)
56 f1of 5457 . . . . . 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 5648 . . 3 (𝜑 → (inr‘𝐴) ∈ ({1o} × V))
60 fvco2 5581 . . 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 5772 . . . 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 5515 . 2 (𝜑 → (𝐺‘(inr‘(inr‘𝐴))) = (𝐺𝐴))
6553, 61, 643eqtrd 2214 1 (𝜑 → (case(𝐹, 𝐺)‘(inr‘𝐴)) = (𝐺𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  Vcvv 2737  cun 3127  cin 3128  c0 3422  {csn 3591  cop 3594   × cxp 4621  ccnv 4622  dom cdm 4623  cima 4626  ccom 4627  Rel wrel 4628  Fun wfun 5206   Fn wfn 5207  wf 5208  1-1-ontowf1o 5211  cfv 5212  1oc1o 6404  inlcinl 7038  inrcinr 7039  casecdjucase 7076
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-suc 4368  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-1st 6135  df-2nd 6136  df-1o 6411  df-inl 7040  df-inr 7041  df-case 7077
This theorem is referenced by:  omp1eomlem  7087
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