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Theorem djufun 7061
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 7018 . . . . 5 (inl ↾ dom 𝐹):dom 𝐹1-1→(dom 𝐹 ⊔ dom 𝐺)
3 df-f1 5188 . . . . . 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 273 . . . . 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 5223 . . . 4 ((Fun 𝐹 ∧ Fun (inl ↾ dom 𝐹)) → Fun (𝐹(inl ↾ dom 𝐹)))
71, 5, 6syl2anc 409 . . 3 (𝜑 → Fun (𝐹(inl ↾ dom 𝐹)))
8 djufun.g . . . 4 (𝜑 → Fun 𝐺)
9 inrresf1 7019 . . . . 5 (inr ↾ dom 𝐺):dom 𝐺1-1→(dom 𝐹 ⊔ dom 𝐺)
10 df-f1 5188 . . . . . 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 273 . . . . 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 5223 . . . 4 ((Fun 𝐺 ∧ Fun (inr ↾ dom 𝐺)) → Fun (𝐺(inr ↾ dom 𝐺)))
148, 12, 13syl2anc 409 . . 3 (𝜑 → Fun (𝐺(inr ↾ dom 𝐺)))
15 dmcoss 4868 . . . . . . 7 dom (𝐹(inl ↾ dom 𝐹)) ⊆ dom (inl ↾ dom 𝐹)
16 df-rn 4610 . . . . . . 7 ran (inl ↾ dom 𝐹) = dom (inl ↾ dom 𝐹)
1715, 16sseqtrri 3173 . . . . . 6 dom (𝐹(inl ↾ dom 𝐹)) ⊆ ran (inl ↾ dom 𝐹)
18 dmcoss 4868 . . . . . . 7 dom (𝐺(inr ↾ dom 𝐺)) ⊆ dom (inr ↾ dom 𝐺)
19 df-rn 4610 . . . . . . 7 ran (inr ↾ dom 𝐺) = dom (inr ↾ dom 𝐺)
2018, 19sseqtrri 3173 . . . . . 6 dom (𝐺(inr ↾ dom 𝐺)) ⊆ ran (inr ↾ dom 𝐺)
21 ss2in 3346 . . . . . 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 423 . . . . 5 (dom (𝐹(inl ↾ dom 𝐹)) ∩ dom (𝐺(inr ↾ dom 𝐺))) ⊆ (ran (inl ↾ dom 𝐹) ∩ ran (inr ↾ dom 𝐺))
23 djuinr 7020 . . . . . 6 (ran (inl ↾ dom 𝐹) ∩ ran (inr ↾ dom 𝐺)) = ∅
2423a1i 9 . . . . 5 (𝜑 → (ran (inl ↾ dom 𝐹) ∩ ran (inr ↾ dom 𝐺)) = ∅)
2522, 24sseqtrid 3188 . . . 4 (𝜑 → (dom (𝐹(inl ↾ dom 𝐹)) ∩ dom (𝐺(inr ↾ dom 𝐺))) ⊆ ∅)
26 ss0 3445 . . . 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 5227 . . 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 1226 . 2 (𝜑 → Fun ((𝐹(inl ↾ dom 𝐹)) ∪ (𝐺(inr ↾ dom 𝐺))))
30 df-djud 7060 . . 3 (𝐹d 𝐺) = ((𝐹(inl ↾ dom 𝐹)) ∪ (𝐺(inr ↾ dom 𝐺)))
3130funeqi 5204 . 2 (Fun (𝐹d 𝐺) ↔ Fun ((𝐹(inl ↾ dom 𝐹)) ∪ (𝐺(inr ↾ dom 𝐺))))
3229, 31sylibr 133 1 (𝜑 → Fun (𝐹d 𝐺))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1342  cun 3110  cin 3111  wss 3112  c0 3405  ccnv 4598  dom cdm 4599  ran crn 4600  cres 4601  ccom 4603  Fun wfun 5177  wf 5179  1-1wf1 5180  cdju 6994  inlcinl 7002  inrcinr 7003  d cdjud 7059
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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4095  ax-nul 4103  ax-pow 4148  ax-pr 4182  ax-un 4406
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-ral 2447  df-rex 2448  df-v 2724  df-sbc 2948  df-dif 3114  df-un 3116  df-in 3118  df-ss 3125  df-nul 3406  df-pw 3556  df-sn 3577  df-pr 3578  df-op 3580  df-uni 3785  df-br 3978  df-opab 4039  df-mpt 4040  df-tr 4076  df-id 4266  df-iord 4339  df-on 4341  df-suc 4344  df-xp 4605  df-rel 4606  df-cnv 4607  df-co 4608  df-dm 4609  df-rn 4610  df-res 4611  df-iota 5148  df-fun 5185  df-fn 5186  df-f 5187  df-f1 5188  df-fo 5189  df-f1o 5190  df-fv 5191  df-1st 6101  df-2nd 6102  df-1o 6376  df-dju 6995  df-inl 7004  df-inr 7005  df-djud 7060
This theorem is referenced by: (None)
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