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Theorem djufun 7170
Description: The "domain-disjoint-union" of two functions is a function. (Contributed by BJ, 10-Jul-2022.)
Hypotheses
Ref Expression
djufun.f (𝜑 → Fun 𝐹)
djufun.g (𝜑 → Fun 𝐺)
Assertion
Ref Expression
djufun (𝜑 → Fun (𝐹d 𝐺))

Proof of Theorem djufun
StepHypRef Expression
1 djufun.f . . . 4 (𝜑 → Fun 𝐹)
2 inlresf1 7127 . . . . 5 (inl ↾ dom 𝐹):dom 𝐹1-1→(dom 𝐹 ⊔ dom 𝐺)
3 df-f1 5263 . . . . . 6 ((inl ↾ dom 𝐹):dom 𝐹1-1→(dom 𝐹 ⊔ dom 𝐺) ↔ ((inl ↾ dom 𝐹):dom 𝐹⟶(dom 𝐹 ⊔ dom 𝐺) ∧ Fun (inl ↾ dom 𝐹)))
43simprbi 275 . . . . 5 ((inl ↾ dom 𝐹):dom 𝐹1-1→(dom 𝐹 ⊔ dom 𝐺) → Fun (inl ↾ dom 𝐹))
52, 4mp1i 10 . . . 4 (𝜑 → Fun (inl ↾ dom 𝐹))
6 funco 5298 . . . 4 ((Fun 𝐹 ∧ Fun (inl ↾ dom 𝐹)) → Fun (𝐹(inl ↾ dom 𝐹)))
71, 5, 6syl2anc 411 . . 3 (𝜑 → Fun (𝐹(inl ↾ dom 𝐹)))
8 djufun.g . . . 4 (𝜑 → Fun 𝐺)
9 inrresf1 7128 . . . . 5 (inr ↾ dom 𝐺):dom 𝐺1-1→(dom 𝐹 ⊔ dom 𝐺)
10 df-f1 5263 . . . . . 6 ((inr ↾ dom 𝐺):dom 𝐺1-1→(dom 𝐹 ⊔ dom 𝐺) ↔ ((inr ↾ dom 𝐺):dom 𝐺⟶(dom 𝐹 ⊔ dom 𝐺) ∧ Fun (inr ↾ dom 𝐺)))
1110simprbi 275 . . . . 5 ((inr ↾ dom 𝐺):dom 𝐺1-1→(dom 𝐹 ⊔ dom 𝐺) → Fun (inr ↾ dom 𝐺))
129, 11mp1i 10 . . . 4 (𝜑 → Fun (inr ↾ dom 𝐺))
13 funco 5298 . . . 4 ((Fun 𝐺 ∧ Fun (inr ↾ dom 𝐺)) → Fun (𝐺(inr ↾ dom 𝐺)))
148, 12, 13syl2anc 411 . . 3 (𝜑 → Fun (𝐺(inr ↾ dom 𝐺)))
15 dmcoss 4935 . . . . . . 7 dom (𝐹(inl ↾ dom 𝐹)) ⊆ dom (inl ↾ dom 𝐹)
16 df-rn 4674 . . . . . . 7 ran (inl ↾ dom 𝐹) = dom (inl ↾ dom 𝐹)
1715, 16sseqtrri 3218 . . . . . 6 dom (𝐹(inl ↾ dom 𝐹)) ⊆ ran (inl ↾ dom 𝐹)
18 dmcoss 4935 . . . . . . 7 dom (𝐺(inr ↾ dom 𝐺)) ⊆ dom (inr ↾ dom 𝐺)
19 df-rn 4674 . . . . . . 7 ran (inr ↾ dom 𝐺) = dom (inr ↾ dom 𝐺)
2018, 19sseqtrri 3218 . . . . . 6 dom (𝐺(inr ↾ dom 𝐺)) ⊆ ran (inr ↾ dom 𝐺)
21 ss2in 3391 . . . . . 6 ((dom (𝐹(inl ↾ dom 𝐹)) ⊆ ran (inl ↾ dom 𝐹) ∧ dom (𝐺(inr ↾ dom 𝐺)) ⊆ ran (inr ↾ dom 𝐺)) → (dom (𝐹(inl ↾ dom 𝐹)) ∩ dom (𝐺(inr ↾ dom 𝐺))) ⊆ (ran (inl ↾ dom 𝐹) ∩ ran (inr ↾ dom 𝐺)))
2217, 20, 21mp2an 426 . . . . 5 (dom (𝐹(inl ↾ dom 𝐹)) ∩ dom (𝐺(inr ↾ dom 𝐺))) ⊆ (ran (inl ↾ dom 𝐹) ∩ ran (inr ↾ dom 𝐺))
23 djuinr 7129 . . . . . 6 (ran (inl ↾ dom 𝐹) ∩ ran (inr ↾ dom 𝐺)) = ∅
2423a1i 9 . . . . 5 (𝜑 → (ran (inl ↾ dom 𝐹) ∩ ran (inr ↾ dom 𝐺)) = ∅)
2522, 24sseqtrid 3233 . . . 4 (𝜑 → (dom (𝐹(inl ↾ dom 𝐹)) ∩ dom (𝐺(inr ↾ dom 𝐺))) ⊆ ∅)
26 ss0 3491 . . . 4 ((dom (𝐹(inl ↾ dom 𝐹)) ∩ dom (𝐺(inr ↾ dom 𝐺))) ⊆ ∅ → (dom (𝐹(inl ↾ dom 𝐹)) ∩ dom (𝐺(inr ↾ dom 𝐺))) = ∅)
2725, 26syl 14 . . 3 (𝜑 → (dom (𝐹(inl ↾ dom 𝐹)) ∩ dom (𝐺(inr ↾ dom 𝐺))) = ∅)
28 funun 5302 . . 3 (((Fun (𝐹(inl ↾ dom 𝐹)) ∧ Fun (𝐺(inr ↾ dom 𝐺))) ∧ (dom (𝐹(inl ↾ dom 𝐹)) ∩ dom (𝐺(inr ↾ dom 𝐺))) = ∅) → Fun ((𝐹(inl ↾ dom 𝐹)) ∪ (𝐺(inr ↾ dom 𝐺))))
297, 14, 27, 28syl21anc 1248 . 2 (𝜑 → Fun ((𝐹(inl ↾ dom 𝐹)) ∪ (𝐺(inr ↾ dom 𝐺))))
30 df-djud 7169 . . 3 (𝐹d 𝐺) = ((𝐹(inl ↾ dom 𝐹)) ∪ (𝐺(inr ↾ dom 𝐺)))
3130funeqi 5279 . 2 (Fun (𝐹d 𝐺) ↔ Fun ((𝐹(inl ↾ dom 𝐹)) ∪ (𝐺(inr ↾ dom 𝐺))))
3229, 31sylibr 134 1 (𝜑 → Fun (𝐹d 𝐺))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  cun 3155  cin 3156  wss 3157  c0 3450  ccnv 4662  dom cdm 4663  ran crn 4664  cres 4665  ccom 4667  Fun wfun 5252  wf 5254  1-1wf1 5255  cdju 7103  inlcinl 7111  inrcinr 7112  d cdjud 7168
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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-suc 4406  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-1st 6198  df-2nd 6199  df-1o 6474  df-dju 7104  df-inl 7113  df-inr 7114  df-djud 7169
This theorem is referenced by: (None)
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