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Theorem casefun 6755
Description: The "case" construction of two functions is a function. (Contributed by BJ, 10-Jul-2022.)
Hypotheses
Ref Expression
casefun.f (𝜑 → Fun 𝐹)
casefun.g (𝜑 → Fun 𝐺)
Assertion
Ref Expression
casefun (𝜑 → Fun case(𝐹, 𝐺))

Proof of Theorem casefun
StepHypRef Expression
1 casefun.f . . . 4 (𝜑 → Fun 𝐹)
2 djulf1o 6729 . . . . . 6 inl:V–1-1-onto→({∅} × V)
3 f1of1 5236 . . . . . 6 (inl:V–1-1-onto→({∅} × V) → inl:V–1-1→({∅} × V))
42, 3ax-mp 7 . . . . 5 inl:V–1-1→({∅} × V)
5 df-f1 5007 . . . . . 6 (inl:V–1-1→({∅} × V) ↔ (inl:V⟶({∅} × V) ∧ Fun inl))
65simprbi 269 . . . . 5 (inl:V–1-1→({∅} × V) → Fun inl)
74, 6mp1i 10 . . . 4 (𝜑 → Fun inl)
8 funco 5040 . . . 4 ((Fun 𝐹 ∧ Fun inl) → Fun (𝐹inl))
91, 7, 8syl2anc 403 . . 3 (𝜑 → Fun (𝐹inl))
10 casefun.g . . . 4 (𝜑 → Fun 𝐺)
11 djurf1o 6730 . . . . . 6 inr:V–1-1-onto→({1𝑜} × V)
12 f1of1 5236 . . . . . 6 (inr:V–1-1-onto→({1𝑜} × V) → inr:V–1-1→({1𝑜} × V))
1311, 12ax-mp 7 . . . . 5 inr:V–1-1→({1𝑜} × V)
14 df-f1 5007 . . . . . 6 (inr:V–1-1→({1𝑜} × V) ↔ (inr:V⟶({1𝑜} × V) ∧ Fun inr))
1514simprbi 269 . . . . 5 (inr:V–1-1→({1𝑜} × V) → Fun inr)
1613, 15mp1i 10 . . . 4 (𝜑 → Fun inr)
17 funco 5040 . . . 4 ((Fun 𝐺 ∧ Fun inr) → Fun (𝐺inr))
1810, 16, 17syl2anc 403 . . 3 (𝜑 → Fun (𝐺inr))
19 dmcoss 4690 . . . . . . 7 dom (𝐹inl) ⊆ dom inl
20 df-rn 4439 . . . . . . 7 ran inl = dom inl
2119, 20sseqtr4i 3057 . . . . . 6 dom (𝐹inl) ⊆ ran inl
22 dmcoss 4690 . . . . . . 7 dom (𝐺inr) ⊆ dom inr
23 df-rn 4439 . . . . . . 7 ran inr = dom inr
2422, 23sseqtr4i 3057 . . . . . 6 dom (𝐺inr) ⊆ ran inr
25 ss2in 3225 . . . . . 6 ((dom (𝐹inl) ⊆ ran inl ∧ dom (𝐺inr) ⊆ ran inr) → (dom (𝐹inl) ∩ dom (𝐺inr)) ⊆ (ran inl ∩ ran inr))
2621, 24, 25mp2an 417 . . . . 5 (dom (𝐹inl) ∩ dom (𝐺inr)) ⊆ (ran inl ∩ ran inr)
27 rnresv 4877 . . . . . . . . 9 ran (inl ↾ V) = ran inl
2827eqcomi 2092 . . . . . . . 8 ran inl = ran (inl ↾ V)
29 rnresv 4877 . . . . . . . . 9 ran (inr ↾ V) = ran inr
3029eqcomi 2092 . . . . . . . 8 ran inr = ran (inr ↾ V)
3128, 30ineq12i 3197 . . . . . . 7 (ran inl ∩ ran inr) = (ran (inl ↾ V) ∩ ran (inr ↾ V))
32 djuinr 6734 . . . . . . 7 (ran (inl ↾ V) ∩ ran (inr ↾ V)) = ∅
3331, 32eqtri 2108 . . . . . 6 (ran inl ∩ ran inr) = ∅
3433a1i 9 . . . . 5 (𝜑 → (ran inl ∩ ran inr) = ∅)
3526, 34syl5sseq 3072 . . . 4 (𝜑 → (dom (𝐹inl) ∩ dom (𝐺inr)) ⊆ ∅)
36 ss0 3320 . . . 4 ((dom (𝐹inl) ∩ dom (𝐺inr)) ⊆ ∅ → (dom (𝐹inl) ∩ dom (𝐺inr)) = ∅)
3735, 36syl 14 . . 3 (𝜑 → (dom (𝐹inl) ∩ dom (𝐺inr)) = ∅)
38 funun 5044 . . 3 (((Fun (𝐹inl) ∧ Fun (𝐺inr)) ∧ (dom (𝐹inl) ∩ dom (𝐺inr)) = ∅) → Fun ((𝐹inl) ∪ (𝐺inr)))
399, 18, 37, 38syl21anc 1173 . 2 (𝜑 → Fun ((𝐹inl) ∪ (𝐺inr)))
40 df-case 6754 . . 3 case(𝐹, 𝐺) = ((𝐹inl) ∪ (𝐺inr))
4140funeqi 5022 . 2 (Fun case(𝐹, 𝐺) ↔ Fun ((𝐹inl) ∪ (𝐺inr)))
4239, 41sylibr 132 1 (𝜑 → Fun case(𝐹, 𝐺))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  Vcvv 2619  cun 2995  cin 2996  wss 2997  c0 3284  {csn 3441   × cxp 4426  ccnv 4427  dom cdm 4428  ran crn 4429  cres 4430  ccom 4432  Fun wfun 4996  wf 4998  1-1wf1 4999  1-1-ontowf1o 5001  1𝑜c1o 6156  inlcinl 6716  inrcinr 6717  casecdjucase 6753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-1st 5893  df-2nd 5894  df-1o 6163  df-inl 6718  df-inr 6719  df-case 6754
This theorem is referenced by:  casef  6758
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