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Theorem caseinj 7088
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 7046 . . . . . . 7 inl = (𝑦 ∈ V ↦ ⟨∅, 𝑦⟩)
21funmpt2 5256 . . . . . 6 Fun inl
3 funcnvcnv 5276 . . . . . 6 (Fun inl → Fun inl)
42, 3ax-mp 5 . . . . 5 Fun inl
5 caseinj.r . . . . 5 (𝜑 → Fun 𝑅)
6 funco 5257 . . . . 5 ((Fun inl ∧ Fun 𝑅) → Fun (inl ∘ 𝑅))
74, 5, 6sylancr 414 . . . 4 (𝜑 → Fun (inl ∘ 𝑅))
8 cnvco 4813 . . . . 5 (𝑅inl) = (inl ∘ 𝑅)
98funeqi 5238 . . . 4 (Fun (𝑅inl) ↔ Fun (inl ∘ 𝑅))
107, 9sylibr 134 . . 3 (𝜑 → Fun (𝑅inl))
11 df-inr 7047 . . . . . . 7 inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
1211funmpt2 5256 . . . . . 6 Fun inr
13 funcnvcnv 5276 . . . . . 6 (Fun inr → Fun inr)
1412, 13ax-mp 5 . . . . 5 Fun inr
15 caseinj.s . . . . 5 (𝜑 → Fun 𝑆)
16 funco 5257 . . . . 5 ((Fun inr ∧ Fun 𝑆) → Fun (inr ∘ 𝑆))
1714, 15, 16sylancr 414 . . . 4 (𝜑 → Fun (inr ∘ 𝑆))
18 cnvco 4813 . . . . 5 (𝑆inr) = (inr ∘ 𝑆)
1918funeqi 5238 . . . 4 (Fun (𝑆inr) ↔ Fun (inr ∘ 𝑆))
2017, 19sylibr 134 . . 3 (𝜑 → Fun (𝑆inr))
21 df-rn 4638 . . . . . . 7 ran (𝑅inl) = dom (𝑅inl)
22 rncoss 4898 . . . . . . 7 ran (𝑅inl) ⊆ ran 𝑅
2321, 22eqsstrri 3189 . . . . . 6 dom (𝑅inl) ⊆ ran 𝑅
24 df-rn 4638 . . . . . . 7 ran (𝑆inr) = dom (𝑆inr)
25 rncoss 4898 . . . . . . 7 ran (𝑆inr) ⊆ ran 𝑆
2624, 25eqsstrri 3189 . . . . . 6 dom (𝑆inr) ⊆ ran 𝑆
27 ss2in 3364 . . . . . 6 ((dom (𝑅inl) ⊆ ran 𝑅 ∧ dom (𝑆inr) ⊆ ran 𝑆) → (dom (𝑅inl) ∩ dom (𝑆inr)) ⊆ (ran 𝑅 ∩ ran 𝑆))
2823, 26, 27mp2an 426 . . . . 5 (dom (𝑅inl) ∩ dom (𝑆inr)) ⊆ (ran 𝑅 ∩ ran 𝑆)
29 caseinj.disj . . . . 5 (𝜑 → (ran 𝑅 ∩ ran 𝑆) = ∅)
3028, 29sseqtrid 3206 . . . 4 (𝜑 → (dom (𝑅inl) ∩ dom (𝑆inr)) ⊆ ∅)
31 ss0 3464 . . . 4 ((dom (𝑅inl) ∩ dom (𝑆inr)) ⊆ ∅ → (dom (𝑅inl) ∩ dom (𝑆inr)) = ∅)
3230, 31syl 14 . . 3 (𝜑 → (dom (𝑅inl) ∩ dom (𝑆inr)) = ∅)
33 funun 5261 . . 3 (((Fun (𝑅inl) ∧ Fun (𝑆inr)) ∧ (dom (𝑅inl) ∩ dom (𝑆inr)) = ∅) → Fun ((𝑅inl) ∪ (𝑆inr)))
3410, 20, 32, 33syl21anc 1237 . 2 (𝜑 → Fun ((𝑅inl) ∪ (𝑆inr)))
35 df-case 7083 . . . . 5 case(𝑅, 𝑆) = ((𝑅inl) ∪ (𝑆inr))
3635cnveqi 4803 . . . 4 case(𝑅, 𝑆) = ((𝑅inl) ∪ (𝑆inr))
37 cnvun 5035 . . . 4 ((𝑅inl) ∪ (𝑆inr)) = ((𝑅inl) ∪ (𝑆inr))
3836, 37eqtri 2198 . . 3 case(𝑅, 𝑆) = ((𝑅inl) ∪ (𝑆inr))
3938funeqi 5238 . 2 (Fun case(𝑅, 𝑆) ↔ Fun ((𝑅inl) ∪ (𝑆inr)))
4034, 39sylibr 134 1 (𝜑 → Fun case(𝑅, 𝑆))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  Vcvv 2738  cun 3128  cin 3129  wss 3130  c0 3423  cop 3596  ccnv 4626  dom cdm 4627  ran crn 4628  ccom 4631  Fun wfun 5211  1oc1o 6410  inlcinl 7044  inrcinr 7045  casecdjucase 7082
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-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  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-ral 2460  df-rex 2461  df-v 2740  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-fun 5219  df-inl 7046  df-inr 7047  df-case 7083
This theorem is referenced by:  casef1  7089
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