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Theorem caseinj 6759
Description: The "case" construction of two injective relations with disjoint ranges is an injective relation. (Contributed by BJ, 10-Jul-2022.)
Hypotheses
Ref Expression
caseinj.r (𝜑 → Fun 𝑅)
caseinj.s (𝜑 → Fun 𝑆)
caseinj.disj (𝜑 → (ran 𝑅 ∩ ran 𝑆) = ∅)
Assertion
Ref Expression
caseinj (𝜑 → Fun case(𝑅, 𝑆))

Proof of Theorem caseinj
StepHypRef Expression
1 df-inl 6718 . . . . . . 7 inl = (𝑦 ∈ V ↦ ⟨∅, 𝑦⟩)
21funmpt2 5039 . . . . . 6 Fun inl
3 funcnvcnv 5059 . . . . . 6 (Fun inl → Fun inl)
42, 3ax-mp 7 . . . . 5 Fun inl
5 caseinj.r . . . . 5 (𝜑 → Fun 𝑅)
6 funco 5040 . . . . 5 ((Fun inl ∧ Fun 𝑅) → Fun (inl ∘ 𝑅))
74, 5, 6sylancr 405 . . . 4 (𝜑 → Fun (inl ∘ 𝑅))
8 cnvco 4609 . . . . 5 (𝑅inl) = (inl ∘ 𝑅)
98funeqi 5022 . . . 4 (Fun (𝑅inl) ↔ Fun (inl ∘ 𝑅))
107, 9sylibr 132 . . 3 (𝜑 → Fun (𝑅inl))
11 df-inr 6719 . . . . . . 7 inr = (𝑥 ∈ V ↦ ⟨1𝑜, 𝑥⟩)
1211funmpt2 5039 . . . . . 6 Fun inr
13 funcnvcnv 5059 . . . . . 6 (Fun inr → Fun inr)
1412, 13ax-mp 7 . . . . 5 Fun inr
15 caseinj.s . . . . 5 (𝜑 → Fun 𝑆)
16 funco 5040 . . . . 5 ((Fun inr ∧ Fun 𝑆) → Fun (inr ∘ 𝑆))
1714, 15, 16sylancr 405 . . . 4 (𝜑 → Fun (inr ∘ 𝑆))
18 cnvco 4609 . . . . 5 (𝑆inr) = (inr ∘ 𝑆)
1918funeqi 5022 . . . 4 (Fun (𝑆inr) ↔ Fun (inr ∘ 𝑆))
2017, 19sylibr 132 . . 3 (𝜑 → Fun (𝑆inr))
21 df-rn 4439 . . . . . . 7 ran (𝑅inl) = dom (𝑅inl)
22 rncoss 4691 . . . . . . 7 ran (𝑅inl) ⊆ ran 𝑅
2321, 22eqsstr3i 3055 . . . . . 6 dom (𝑅inl) ⊆ ran 𝑅
24 df-rn 4439 . . . . . . 7 ran (𝑆inr) = dom (𝑆inr)
25 rncoss 4691 . . . . . . 7 ran (𝑆inr) ⊆ ran 𝑆
2624, 25eqsstr3i 3055 . . . . . 6 dom (𝑆inr) ⊆ ran 𝑆
27 ss2in 3225 . . . . . 6 ((dom (𝑅inl) ⊆ ran 𝑅 ∧ dom (𝑆inr) ⊆ ran 𝑆) → (dom (𝑅inl) ∩ dom (𝑆inr)) ⊆ (ran 𝑅 ∩ ran 𝑆))
2823, 26, 27mp2an 417 . . . . 5 (dom (𝑅inl) ∩ dom (𝑆inr)) ⊆ (ran 𝑅 ∩ ran 𝑆)
29 caseinj.disj . . . . 5 (𝜑 → (ran 𝑅 ∩ ran 𝑆) = ∅)
3028, 29syl5sseq 3072 . . . 4 (𝜑 → (dom (𝑅inl) ∩ dom (𝑆inr)) ⊆ ∅)
31 ss0 3320 . . . 4 ((dom (𝑅inl) ∩ dom (𝑆inr)) ⊆ ∅ → (dom (𝑅inl) ∩ dom (𝑆inr)) = ∅)
3230, 31syl 14 . . 3 (𝜑 → (dom (𝑅inl) ∩ dom (𝑆inr)) = ∅)
33 funun 5044 . . 3 (((Fun (𝑅inl) ∧ Fun (𝑆inr)) ∧ (dom (𝑅inl) ∩ dom (𝑆inr)) = ∅) → Fun ((𝑅inl) ∪ (𝑆inr)))
3410, 20, 32, 33syl21anc 1173 . 2 (𝜑 → Fun ((𝑅inl) ∪ (𝑆inr)))
35 df-case 6754 . . . . 5 case(𝑅, 𝑆) = ((𝑅inl) ∪ (𝑆inr))
3635cnveqi 4599 . . . 4 case(𝑅, 𝑆) = ((𝑅inl) ∪ (𝑆inr))
37 cnvun 4824 . . . 4 ((𝑅inl) ∪ (𝑆inr)) = ((𝑅inl) ∪ (𝑆inr))
3836, 37eqtri 2108 . . 3 case(𝑅, 𝑆) = ((𝑅inl) ∪ (𝑆inr))
3938funeqi 5022 . 2 (Fun case(𝑅, 𝑆) ↔ Fun ((𝑅inl) ∪ (𝑆inr)))
4034, 39sylibr 132 1 (𝜑 → Fun case(𝑅, 𝑆))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  Vcvv 2619  cun 2995  cin 2996  wss 2997  c0 3284  cop 3444  ccnv 4427  dom cdm 4428  ran crn 4429  ccom 4432  Fun wfun 4996  1𝑜c1o 6156  inlcinl 6716  inrcinr 6717  casecdjucase 6753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-fun 5004  df-inl 6718  df-inr 6719  df-case 6754
This theorem is referenced by:  casef1  6760
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