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Theorem caseinl 6976
 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 6969 . . . 4 case(𝐹, 𝐺) = ((𝐹inl) ∪ (𝐺inr))
21fveq1i 5422 . . 3 (case(𝐹, 𝐺)‘(inl‘𝐴)) = (((𝐹inl) ∪ (𝐺inr))‘(inl‘𝐴))
3 caseinl.f . . . . . . 7 (𝜑𝐹 Fn 𝐵)
4 fnfun 5220 . . . . . . 7 (𝐹 Fn 𝐵 → Fun 𝐹)
53, 4syl 14 . . . . . 6 (𝜑 → Fun 𝐹)
6 djulf1o 6943 . . . . . . . 8 inl:V–1-1-onto→({∅} × V)
7 f1ocnv 5380 . . . . . . . 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 5369 . . . . . . 7 (inl:({∅} × V)–1-1-onto→V → Fun inl)
108, 9ax-mp 5 . . . . . 6 Fun inl
11 funco 5163 . . . . . 6 ((Fun 𝐹 ∧ Fun inl) → Fun (𝐹inl))
125, 10, 11sylancl 409 . . . . 5 (𝜑 → Fun (𝐹inl))
13 dmco 5047 . . . . . 6 dom (𝐹inl) = (inl “ dom 𝐹)
14 df-inl 6932 . . . . . . . . . . 11 inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
1514funmpt2 5162 . . . . . . . . . 10 Fun inl
16 funrel 5140 . . . . . . . . . 10 (Fun inl → Rel inl)
1715, 16ax-mp 5 . . . . . . . . 9 Rel inl
18 dfrel2 4989 . . . . . . . . 9 (Rel inl ↔ inl = inl)
1917, 18mpbi 144 . . . . . . . 8 inl = inl
2019a1i 9 . . . . . . 7 (𝜑inl = inl)
21 fndm 5222 . . . . . . . 8 (𝐹 Fn 𝐵 → dom 𝐹 = 𝐵)
223, 21syl 14 . . . . . . 7 (𝜑 → dom 𝐹 = 𝐵)
2320, 22imaeq12d 4882 . . . . . 6 (𝜑 → (inl “ dom 𝐹) = (inl “ 𝐵))
2413, 23syl5eq 2184 . . . . 5 (𝜑 → dom (𝐹inl) = (inl “ 𝐵))
25 df-fn 5126 . . . . 5 ((𝐹inl) Fn (inl “ 𝐵) ↔ (Fun (𝐹inl) ∧ dom (𝐹inl) = (inl “ 𝐵)))
2612, 24, 25sylanbrc 413 . . . 4 (𝜑 → (𝐹inl) Fn (inl “ 𝐵))
27 caseinl.g . . . . . 6 (𝜑 → Fun 𝐺)
28 djurf1o 6944 . . . . . . . 8 inr:V–1-1-onto→({1o} × V)
29 f1ocnv 5380 . . . . . . . 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 5369 . . . . . . 7 (inr:({1o} × V)–1-1-onto→V → Fun inr)
3230, 31ax-mp 5 . . . . . 6 Fun inr
33 funco 5163 . . . . . 6 ((Fun 𝐺 ∧ Fun inr) → Fun (𝐺inr))
3427, 32, 33sylancl 409 . . . . 5 (𝜑 → Fun (𝐺inr))
35 dmco 5047 . . . . . . 7 dom (𝐺inr) = (inr “ dom 𝐺)
36 imacnvcnv 5003 . . . . . . 7 (inr “ dom 𝐺) = (inr “ dom 𝐺)
3735, 36eqtri 2160 . . . . . 6 dom (𝐺inr) = (inr “ dom 𝐺)
3837a1i 9 . . . . 5 (𝜑 → dom (𝐺inr) = (inr “ dom 𝐺))
39 df-fn 5126 . . . . 5 ((𝐺inr) Fn (inr “ dom 𝐺) ↔ (Fun (𝐺inr) ∧ dom (𝐺inr) = (inr “ dom 𝐺)))
4034, 38, 39sylanbrc 413 . . . 4 (𝜑 → (𝐺inr) Fn (inr “ dom 𝐺))
41 djuin 6949 . . . . 5 ((inl “ 𝐵) ∩ (inr “ dom 𝐺)) = ∅
4241a1i 9 . . . 4 (𝜑 → ((inl “ 𝐵) ∩ (inr “ dom 𝐺)) = ∅)
43 caseinl.a . . . . . . . 8 (𝜑𝐴𝐵)
4443elexd 2699 . . . . . . 7 (𝜑𝐴 ∈ V)
45 f1odm 5371 . . . . . . . 8 (inl:V–1-1-onto→({∅} × V) → dom inl = V)
466, 45ax-mp 5 . . . . . . 7 dom inl = V
4744, 46eleqtrrdi 2233 . . . . . 6 (𝜑𝐴 ∈ dom inl)
4847, 15jctil 310 . . . . 5 (𝜑 → (Fun inl ∧ 𝐴 ∈ dom inl))
49 funfvima 5649 . . . . 5 ((Fun inl ∧ 𝐴 ∈ dom inl) → (𝐴𝐵 → (inl‘𝐴) ∈ (inl “ 𝐵)))
5048, 43, 49sylc 62 . . . 4 (𝜑 → (inl‘𝐴) ∈ (inl “ 𝐵))
51 fvun1 5487 . . . 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 1220 . . 3 (𝜑 → (((𝐹inl) ∪ (𝐺inr))‘(inl‘𝐴)) = ((𝐹inl)‘(inl‘𝐴)))
532, 52syl5eq 2184 . 2 (𝜑 → (case(𝐹, 𝐺)‘(inl‘𝐴)) = ((𝐹inl)‘(inl‘𝐴)))
54 f1ofn 5368 . . . 4 (inl:({∅} × V)–1-1-onto→V → inl Fn ({∅} × V))
558, 54ax-mp 5 . . 3 inl Fn ({∅} × V)
56 f1of 5367 . . . . . 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, 44ffvelrnd 5556 . . 3 (𝜑 → (inl‘𝐴) ∈ ({∅} × V))
60 fvco2 5490 . . 3 ((inl Fn ({∅} × V) ∧ (inl‘𝐴) ∈ ({∅} × V)) → ((𝐹inl)‘(inl‘𝐴)) = (𝐹‘(inl‘(inl‘𝐴))))
6155, 59, 60sylancr 410 . 2 (𝜑 → ((𝐹inl)‘(inl‘𝐴)) = (𝐹‘(inl‘(inl‘𝐴))))
62 f1ocnvfv1 5678 . . . 4 ((inl:V–1-1-onto→({∅} × V) ∧ 𝐴 ∈ V) → (inl‘(inl‘𝐴)) = 𝐴)
636, 44, 62sylancr 410 . . 3 (𝜑 → (inl‘(inl‘𝐴)) = 𝐴)
6463fveq2d 5425 . 2 (𝜑 → (𝐹‘(inl‘(inl‘𝐴))) = (𝐹𝐴))
6553, 61, 643eqtrd 2176 1 (𝜑 → (case(𝐹, 𝐺)‘(inl‘𝐴)) = (𝐹𝐴))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331   ∈ wcel 1480  Vcvv 2686   ∪ cun 3069   ∩ cin 3070  ∅c0 3363  {csn 3527  ⟨cop 3530   × cxp 4537  ◡ccnv 4538  dom cdm 4539   “ cima 4542   ∘ ccom 4543  Rel wrel 4544  Fun wfun 5117   Fn wfn 5118  ⟶wf 5119  –1-1-onto→wf1o 5122  ‘cfv 5123  1oc1o 6306  inlcinl 6930  inrcinr 6931  casecdjucase 6968 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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-1st 6038  df-2nd 6039  df-1o 6313  df-inl 6932  df-inr 6933  df-case 6969 This theorem is referenced by:  omp1eomlem  6979  ctm  6994
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