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Theorem caseinl 6862
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 6855 . . . 4 case(𝐹, 𝐺) = ((𝐹inl) ∪ (𝐺inr))
21fveq1i 5341 . . 3 (case(𝐹, 𝐺)‘(inl‘𝐴)) = (((𝐹inl) ∪ (𝐺inr))‘(inl‘𝐴))
3 caseinl.f . . . . . . 7 (𝜑𝐹 Fn 𝐵)
4 fnfun 5145 . . . . . . 7 (𝐹 Fn 𝐵 → Fun 𝐹)
53, 4syl 14 . . . . . 6 (𝜑 → Fun 𝐹)
6 djulf1o 6830 . . . . . . . 8 inl:V–1-1-onto→({∅} × V)
7 f1ocnv 5301 . . . . . . . 8 (inl:V–1-1-onto→({∅} × V) → inl:({∅} × V)–1-1-onto→V)
86, 7ax-mp 7 . . . . . . 7 inl:({∅} × V)–1-1-onto→V
9 f1ofun 5290 . . . . . . 7 (inl:({∅} × V)–1-1-onto→V → Fun inl)
108, 9ax-mp 7 . . . . . 6 Fun inl
11 funco 5088 . . . . . 6 ((Fun 𝐹 ∧ Fun inl) → Fun (𝐹inl))
125, 10, 11sylancl 405 . . . . 5 (𝜑 → Fun (𝐹inl))
13 dmco 4973 . . . . . 6 dom (𝐹inl) = (inl “ dom 𝐹)
14 df-inl 6819 . . . . . . . . . . 11 inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
1514funmpt2 5087 . . . . . . . . . 10 Fun inl
16 funrel 5066 . . . . . . . . . 10 (Fun inl → Rel inl)
1715, 16ax-mp 7 . . . . . . . . 9 Rel inl
18 dfrel2 4915 . . . . . . . . 9 (Rel inl ↔ inl = inl)
1917, 18mpbi 144 . . . . . . . 8 inl = inl
2019a1i 9 . . . . . . 7 (𝜑inl = inl)
21 fndm 5147 . . . . . . . 8 (𝐹 Fn 𝐵 → dom 𝐹 = 𝐵)
223, 21syl 14 . . . . . . 7 (𝜑 → dom 𝐹 = 𝐵)
2320, 22imaeq12d 4808 . . . . . 6 (𝜑 → (inl “ dom 𝐹) = (inl “ 𝐵))
2413, 23syl5eq 2139 . . . . 5 (𝜑 → dom (𝐹inl) = (inl “ 𝐵))
25 df-fn 5052 . . . . 5 ((𝐹inl) Fn (inl “ 𝐵) ↔ (Fun (𝐹inl) ∧ dom (𝐹inl) = (inl “ 𝐵)))
2612, 24, 25sylanbrc 409 . . . 4 (𝜑 → (𝐹inl) Fn (inl “ 𝐵))
27 caseinl.g . . . . . 6 (𝜑 → Fun 𝐺)
28 djurf1o 6831 . . . . . . . 8 inr:V–1-1-onto→({1o} × V)
29 f1ocnv 5301 . . . . . . . 8 (inr:V–1-1-onto→({1o} × V) → inr:({1o} × V)–1-1-onto→V)
3028, 29ax-mp 7 . . . . . . 7 inr:({1o} × V)–1-1-onto→V
31 f1ofun 5290 . . . . . . 7 (inr:({1o} × V)–1-1-onto→V → Fun inr)
3230, 31ax-mp 7 . . . . . 6 Fun inr
33 funco 5088 . . . . . 6 ((Fun 𝐺 ∧ Fun inr) → Fun (𝐺inr))
3427, 32, 33sylancl 405 . . . . 5 (𝜑 → Fun (𝐺inr))
35 dmco 4973 . . . . . . 7 dom (𝐺inr) = (inr “ dom 𝐺)
36 imacnvcnv 4929 . . . . . . 7 (inr “ dom 𝐺) = (inr “ dom 𝐺)
3735, 36eqtri 2115 . . . . . 6 dom (𝐺inr) = (inr “ dom 𝐺)
3837a1i 9 . . . . 5 (𝜑 → dom (𝐺inr) = (inr “ dom 𝐺))
39 df-fn 5052 . . . . 5 ((𝐺inr) Fn (inr “ dom 𝐺) ↔ (Fun (𝐺inr) ∧ dom (𝐺inr) = (inr “ dom 𝐺)))
4034, 38, 39sylanbrc 409 . . . 4 (𝜑 → (𝐺inr) Fn (inr “ dom 𝐺))
41 djuin 6836 . . . . 5 ((inl “ 𝐵) ∩ (inr “ dom 𝐺)) = ∅
4241a1i 9 . . . 4 (𝜑 → ((inl “ 𝐵) ∩ (inr “ dom 𝐺)) = ∅)
43 caseinl.a . . . . . . . 8 (𝜑𝐴𝐵)
4443elexd 2646 . . . . . . 7 (𝜑𝐴 ∈ V)
45 f1odm 5292 . . . . . . . 8 (inl:V–1-1-onto→({∅} × V) → dom inl = V)
466, 45ax-mp 7 . . . . . . 7 dom inl = V
4744, 46syl6eleqr 2188 . . . . . 6 (𝜑𝐴 ∈ dom inl)
4847, 15jctil 306 . . . . 5 (𝜑 → (Fun inl ∧ 𝐴 ∈ dom inl))
49 funfvima 5565 . . . . 5 ((Fun inl ∧ 𝐴 ∈ dom inl) → (𝐴𝐵 → (inl‘𝐴) ∈ (inl “ 𝐵)))
5048, 43, 49sylc 62 . . . 4 (𝜑 → (inl‘𝐴) ∈ (inl “ 𝐵))
51 fvun1 5405 . . . 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 1185 . . 3 (𝜑 → (((𝐹inl) ∪ (𝐺inr))‘(inl‘𝐴)) = ((𝐹inl)‘(inl‘𝐴)))
532, 52syl5eq 2139 . 2 (𝜑 → (case(𝐹, 𝐺)‘(inl‘𝐴)) = ((𝐹inl)‘(inl‘𝐴)))
54 f1ofn 5289 . . . 4 (inl:({∅} × V)–1-1-onto→V → inl Fn ({∅} × V))
558, 54ax-mp 7 . . 3 inl Fn ({∅} × V)
56 f1of 5288 . . . . . 6 (inl:V–1-1-onto→({∅} × V) → inl:V⟶({∅} × V))
576, 56ax-mp 7 . . . . 5 inl:V⟶({∅} × V)
5857a1i 9 . . . 4 (𝜑 → inl:V⟶({∅} × V))
5958, 44ffvelrnd 5474 . . 3 (𝜑 → (inl‘𝐴) ∈ ({∅} × V))
60 fvco2 5408 . . 3 ((inl Fn ({∅} × V) ∧ (inl‘𝐴) ∈ ({∅} × V)) → ((𝐹inl)‘(inl‘𝐴)) = (𝐹‘(inl‘(inl‘𝐴))))
6155, 59, 60sylancr 406 . 2 (𝜑 → ((𝐹inl)‘(inl‘𝐴)) = (𝐹‘(inl‘(inl‘𝐴))))
62 f1ocnvfv1 5594 . . . 4 ((inl:V–1-1-onto→({∅} × V) ∧ 𝐴 ∈ V) → (inl‘(inl‘𝐴)) = 𝐴)
636, 44, 62sylancr 406 . . 3 (𝜑 → (inl‘(inl‘𝐴)) = 𝐴)
6463fveq2d 5344 . 2 (𝜑 → (𝐹‘(inl‘(inl‘𝐴))) = (𝐹𝐴))
6553, 61, 643eqtrd 2131 1 (𝜑 → (case(𝐹, 𝐺)‘(inl‘𝐴)) = (𝐹𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1296  wcel 1445  Vcvv 2633  cun 3011  cin 3012  c0 3302  {csn 3466  cop 3469   × cxp 4465  ccnv 4466  dom cdm 4467  cima 4470  ccom 4471  Rel wrel 4472  Fun wfun 5043   Fn wfn 5044  wf 5045  1-1-ontowf1o 5048  cfv 5049  1oc1o 6212  inlcinl 6817  inrcinr 6818  casecdjucase 6854
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 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-id 4144  df-iord 4217  df-on 4219  df-suc 4222  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-1st 5949  df-2nd 5950  df-1o 6219  df-inl 6819  df-inr 6820  df-case 6855
This theorem is referenced by:  ctm  6871
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