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Theorem djufun 7105
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 7062 . . . . 5 (inl ↾ dom 𝐹):dom 𝐹1-1→(dom 𝐹 ⊔ dom 𝐺)
3 df-f1 5223 . . . . . 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 5258 . . . 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 7063 . . . . 5 (inr ↾ dom 𝐺):dom 𝐺1-1→(dom 𝐹 ⊔ dom 𝐺)
10 df-f1 5223 . . . . . 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 5258 . . . 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 4898 . . . . . . 7 dom (𝐹(inl ↾ dom 𝐹)) ⊆ dom (inl ↾ dom 𝐹)
16 df-rn 4639 . . . . . . 7 ran (inl ↾ dom 𝐹) = dom (inl ↾ dom 𝐹)
1715, 16sseqtrri 3192 . . . . . 6 dom (𝐹(inl ↾ dom 𝐹)) ⊆ ran (inl ↾ dom 𝐹)
18 dmcoss 4898 . . . . . . 7 dom (𝐺(inr ↾ dom 𝐺)) ⊆ dom (inr ↾ dom 𝐺)
19 df-rn 4639 . . . . . . 7 ran (inr ↾ dom 𝐺) = dom (inr ↾ dom 𝐺)
2018, 19sseqtrri 3192 . . . . . 6 dom (𝐺(inr ↾ dom 𝐺)) ⊆ ran (inr ↾ dom 𝐺)
21 ss2in 3365 . . . . . 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 7064 . . . . . 6 (ran (inl ↾ dom 𝐹) ∩ ran (inr ↾ dom 𝐺)) = ∅
2423a1i 9 . . . . 5 (𝜑 → (ran (inl ↾ dom 𝐹) ∩ ran (inr ↾ dom 𝐺)) = ∅)
2522, 24sseqtrid 3207 . . . 4 (𝜑 → (dom (𝐹(inl ↾ dom 𝐹)) ∩ dom (𝐺(inr ↾ dom 𝐺))) ⊆ ∅)
26 ss0 3465 . . . 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 5262 . . 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 1237 . 2 (𝜑 → Fun ((𝐹(inl ↾ dom 𝐹)) ∪ (𝐺(inr ↾ dom 𝐺))))
30 df-djud 7104 . . 3 (𝐹d 𝐺) = ((𝐹(inl ↾ dom 𝐹)) ∪ (𝐺(inr ↾ dom 𝐺)))
3130funeqi 5239 . 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 1353  cun 3129  cin 3130  wss 3131  c0 3424  ccnv 4627  dom cdm 4628  ran crn 4629  cres 4630  ccom 4632  Fun wfun 5212  wf 5214  1-1wf1 5215  cdju 7038  inlcinl 7046  inrcinr 7047  d cdjud 7103
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-1st 6143  df-2nd 6144  df-1o 6419  df-dju 7039  df-inl 7048  df-inr 7049  df-djud 7104
This theorem is referenced by: (None)
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