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Theorem djufun 6989
 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 6946 . . . . 5 (inl ↾ dom 𝐹):dom 𝐹1-1→(dom 𝐹 ⊔ dom 𝐺)
3 df-f1 5128 . . . . . 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 5163 . . . 4 ((Fun 𝐹 ∧ Fun (inl ↾ dom 𝐹)) → Fun (𝐹(inl ↾ dom 𝐹)))
71, 5, 6syl2anc 408 . . 3 (𝜑 → Fun (𝐹(inl ↾ dom 𝐹)))
8 djufun.g . . . 4 (𝜑 → Fun 𝐺)
9 inrresf1 6947 . . . . 5 (inr ↾ dom 𝐺):dom 𝐺1-1→(dom 𝐹 ⊔ dom 𝐺)
10 df-f1 5128 . . . . . 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 5163 . . . 4 ((Fun 𝐺 ∧ Fun (inr ↾ dom 𝐺)) → Fun (𝐺(inr ↾ dom 𝐺)))
148, 12, 13syl2anc 408 . . 3 (𝜑 → Fun (𝐺(inr ↾ dom 𝐺)))
15 dmcoss 4808 . . . . . . 7 dom (𝐹(inl ↾ dom 𝐹)) ⊆ dom (inl ↾ dom 𝐹)
16 df-rn 4550 . . . . . . 7 ran (inl ↾ dom 𝐹) = dom (inl ↾ dom 𝐹)
1715, 16sseqtrri 3132 . . . . . 6 dom (𝐹(inl ↾ dom 𝐹)) ⊆ ran (inl ↾ dom 𝐹)
18 dmcoss 4808 . . . . . . 7 dom (𝐺(inr ↾ dom 𝐺)) ⊆ dom (inr ↾ dom 𝐺)
19 df-rn 4550 . . . . . . 7 ran (inr ↾ dom 𝐺) = dom (inr ↾ dom 𝐺)
2018, 19sseqtrri 3132 . . . . . 6 dom (𝐺(inr ↾ dom 𝐺)) ⊆ ran (inr ↾ dom 𝐺)
21 ss2in 3304 . . . . . 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 422 . . . . 5 (dom (𝐹(inl ↾ dom 𝐹)) ∩ dom (𝐺(inr ↾ dom 𝐺))) ⊆ (ran (inl ↾ dom 𝐹) ∩ ran (inr ↾ dom 𝐺))
23 djuinr 6948 . . . . . 6 (ran (inl ↾ dom 𝐹) ∩ ran (inr ↾ dom 𝐺)) = ∅
2423a1i 9 . . . . 5 (𝜑 → (ran (inl ↾ dom 𝐹) ∩ ran (inr ↾ dom 𝐺)) = ∅)
2522, 24sseqtrid 3147 . . . 4 (𝜑 → (dom (𝐹(inl ↾ dom 𝐹)) ∩ dom (𝐺(inr ↾ dom 𝐺))) ⊆ ∅)
26 ss0 3403 . . . 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 5167 . . 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 1215 . 2 (𝜑 → Fun ((𝐹(inl ↾ dom 𝐹)) ∪ (𝐺(inr ↾ dom 𝐺))))
30 df-djud 6988 . . 3 (𝐹d 𝐺) = ((𝐹(inl ↾ dom 𝐹)) ∪ (𝐺(inr ↾ dom 𝐺)))
3130funeqi 5144 . 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 1331   ∪ cun 3069   ∩ cin 3070   ⊆ wss 3071  ∅c0 3363  ◡ccnv 4538  dom cdm 4539  ran crn 4540   ↾ cres 4541   ∘ ccom 4543  Fun wfun 5117  ⟶wf 5119  –1-1→wf1 5120   ⊔ cdju 6922  inlcinl 6930  inrcinr 6931   ⊔d cdjud 6987 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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-1st 6038  df-2nd 6039  df-1o 6313  df-dju 6923  df-inl 6932  df-inr 6933  df-djud 6988 This theorem is referenced by: (None)
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