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Theorem casefun 6936
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 6909 . . . . . 6 inl:V–1-1-onto→({∅} × V)
3 f1of1 5332 . . . . . 6 (inl:V–1-1-onto→({∅} × V) → inl:V–1-1→({∅} × V))
42, 3ax-mp 5 . . . . 5 inl:V–1-1→({∅} × V)
5 df-f1 5096 . . . . . 6 (inl:V–1-1→({∅} × V) ↔ (inl:V⟶({∅} × V) ∧ Fun inl))
65simprbi 271 . . . . 5 (inl:V–1-1→({∅} × V) → Fun inl)
74, 6mp1i 10 . . . 4 (𝜑 → Fun inl)
8 funco 5131 . . . 4 ((Fun 𝐹 ∧ Fun inl) → Fun (𝐹inl))
91, 7, 8syl2anc 406 . . 3 (𝜑 → Fun (𝐹inl))
10 casefun.g . . . 4 (𝜑 → Fun 𝐺)
11 djurf1o 6910 . . . . . 6 inr:V–1-1-onto→({1o} × V)
12 f1of1 5332 . . . . . 6 (inr:V–1-1-onto→({1o} × V) → inr:V–1-1→({1o} × V))
1311, 12ax-mp 5 . . . . 5 inr:V–1-1→({1o} × V)
14 df-f1 5096 . . . . . 6 (inr:V–1-1→({1o} × V) ↔ (inr:V⟶({1o} × V) ∧ Fun inr))
1514simprbi 271 . . . . 5 (inr:V–1-1→({1o} × V) → Fun inr)
1613, 15mp1i 10 . . . 4 (𝜑 → Fun inr)
17 funco 5131 . . . 4 ((Fun 𝐺 ∧ Fun inr) → Fun (𝐺inr))
1810, 16, 17syl2anc 406 . . 3 (𝜑 → Fun (𝐺inr))
19 dmcoss 4776 . . . . . . 7 dom (𝐹inl) ⊆ dom inl
20 df-rn 4518 . . . . . . 7 ran inl = dom inl
2119, 20sseqtrri 3100 . . . . . 6 dom (𝐹inl) ⊆ ran inl
22 dmcoss 4776 . . . . . . 7 dom (𝐺inr) ⊆ dom inr
23 df-rn 4518 . . . . . . 7 ran inr = dom inr
2422, 23sseqtrri 3100 . . . . . 6 dom (𝐺inr) ⊆ ran inr
25 ss2in 3272 . . . . . 6 ((dom (𝐹inl) ⊆ ran inl ∧ dom (𝐺inr) ⊆ ran inr) → (dom (𝐹inl) ∩ dom (𝐺inr)) ⊆ (ran inl ∩ ran inr))
2621, 24, 25mp2an 420 . . . . 5 (dom (𝐹inl) ∩ dom (𝐺inr)) ⊆ (ran inl ∩ ran inr)
27 rnresv 4966 . . . . . . . . 9 ran (inl ↾ V) = ran inl
2827eqcomi 2119 . . . . . . . 8 ran inl = ran (inl ↾ V)
29 rnresv 4966 . . . . . . . . 9 ran (inr ↾ V) = ran inr
3029eqcomi 2119 . . . . . . . 8 ran inr = ran (inr ↾ V)
3128, 30ineq12i 3243 . . . . . . 7 (ran inl ∩ ran inr) = (ran (inl ↾ V) ∩ ran (inr ↾ V))
32 djuinr 6914 . . . . . . 7 (ran (inl ↾ V) ∩ ran (inr ↾ V)) = ∅
3331, 32eqtri 2136 . . . . . 6 (ran inl ∩ ran inr) = ∅
3433a1i 9 . . . . 5 (𝜑 → (ran inl ∩ ran inr) = ∅)
3526, 34sseqtrid 3115 . . . 4 (𝜑 → (dom (𝐹inl) ∩ dom (𝐺inr)) ⊆ ∅)
36 ss0 3371 . . . 4 ((dom (𝐹inl) ∩ dom (𝐺inr)) ⊆ ∅ → (dom (𝐹inl) ∩ dom (𝐺inr)) = ∅)
3735, 36syl 14 . . 3 (𝜑 → (dom (𝐹inl) ∩ dom (𝐺inr)) = ∅)
38 funun 5135 . . 3 (((Fun (𝐹inl) ∧ Fun (𝐺inr)) ∧ (dom (𝐹inl) ∩ dom (𝐺inr)) = ∅) → Fun ((𝐹inl) ∪ (𝐺inr)))
399, 18, 37, 38syl21anc 1198 . 2 (𝜑 → Fun ((𝐹inl) ∪ (𝐺inr)))
40 df-case 6935 . . 3 case(𝐹, 𝐺) = ((𝐹inl) ∪ (𝐺inr))
4140funeqi 5112 . 2 (Fun case(𝐹, 𝐺) ↔ Fun ((𝐹inl) ∪ (𝐺inr)))
4239, 41sylibr 133 1 (𝜑 → Fun case(𝐹, 𝐺))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1314  Vcvv 2658  cun 3037  cin 3038  wss 3039  c0 3331  {csn 3495   × cxp 4505  ccnv 4506  dom cdm 4507  ran crn 4508  cres 4509  ccom 4511  Fun wfun 5085  wf 5087  1-1wf1 5088  1-1-ontowf1o 5090  1oc1o 6272  inlcinl 6896  inrcinr 6897  casecdjucase 6934
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-suc 4261  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-1st 6004  df-2nd 6005  df-1o 6279  df-inl 6898  df-inr 6899  df-case 6935
This theorem is referenced by:  casef  6939
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