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Theorem caseinj 7087
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 7045 . . . . . . 7 inl = (𝑦 ∈ V ↦ ⟨∅, 𝑦⟩)
21funmpt2 5255 . . . . . 6 Fun inl
3 funcnvcnv 5275 . . . . . 6 (Fun inl → Fun inl)
42, 3ax-mp 5 . . . . 5 Fun inl
5 caseinj.r . . . . 5 (𝜑 → Fun 𝑅)
6 funco 5256 . . . . 5 ((Fun inl ∧ Fun 𝑅) → Fun (inl ∘ 𝑅))
74, 5, 6sylancr 414 . . . 4 (𝜑 → Fun (inl ∘ 𝑅))
8 cnvco 4812 . . . . 5 (𝑅inl) = (inl ∘ 𝑅)
98funeqi 5237 . . . 4 (Fun (𝑅inl) ↔ Fun (inl ∘ 𝑅))
107, 9sylibr 134 . . 3 (𝜑 → Fun (𝑅inl))
11 df-inr 7046 . . . . . . 7 inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
1211funmpt2 5255 . . . . . 6 Fun inr
13 funcnvcnv 5275 . . . . . 6 (Fun inr → Fun inr)
1412, 13ax-mp 5 . . . . 5 Fun inr
15 caseinj.s . . . . 5 (𝜑 → Fun 𝑆)
16 funco 5256 . . . . 5 ((Fun inr ∧ Fun 𝑆) → Fun (inr ∘ 𝑆))
1714, 15, 16sylancr 414 . . . 4 (𝜑 → Fun (inr ∘ 𝑆))
18 cnvco 4812 . . . . 5 (𝑆inr) = (inr ∘ 𝑆)
1918funeqi 5237 . . . 4 (Fun (𝑆inr) ↔ Fun (inr ∘ 𝑆))
2017, 19sylibr 134 . . 3 (𝜑 → Fun (𝑆inr))
21 df-rn 4637 . . . . . . 7 ran (𝑅inl) = dom (𝑅inl)
22 rncoss 4897 . . . . . . 7 ran (𝑅inl) ⊆ ran 𝑅
2321, 22eqsstrri 3188 . . . . . 6 dom (𝑅inl) ⊆ ran 𝑅
24 df-rn 4637 . . . . . . 7 ran (𝑆inr) = dom (𝑆inr)
25 rncoss 4897 . . . . . . 7 ran (𝑆inr) ⊆ ran 𝑆
2624, 25eqsstrri 3188 . . . . . 6 dom (𝑆inr) ⊆ ran 𝑆
27 ss2in 3363 . . . . . 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 3205 . . . 4 (𝜑 → (dom (𝑅inl) ∩ dom (𝑆inr)) ⊆ ∅)
31 ss0 3463 . . . 4 ((dom (𝑅inl) ∩ dom (𝑆inr)) ⊆ ∅ → (dom (𝑅inl) ∩ dom (𝑆inr)) = ∅)
3230, 31syl 14 . . 3 (𝜑 → (dom (𝑅inl) ∩ dom (𝑆inr)) = ∅)
33 funun 5260 . . 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 7082 . . . . 5 case(𝑅, 𝑆) = ((𝑅inl) ∪ (𝑆inr))
3635cnveqi 4802 . . . 4 case(𝑅, 𝑆) = ((𝑅inl) ∪ (𝑆inr))
37 cnvun 5034 . . . 4 ((𝑅inl) ∪ (𝑆inr)) = ((𝑅inl) ∪ (𝑆inr))
3836, 37eqtri 2198 . . 3 case(𝑅, 𝑆) = ((𝑅inl) ∪ (𝑆inr))
3938funeqi 5237 . 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 2737  cun 3127  cin 3128  wss 3129  c0 3422  cop 3595  ccnv 4625  dom cdm 4626  ran crn 4627  ccom 4630  Fun wfun 5210  1oc1o 6409  inlcinl 7043  inrcinr 7044  casecdjucase 7081
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 4121  ax-pow 4174  ax-pr 4209
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 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-fun 5218  df-inl 7045  df-inr 7046  df-case 7082
This theorem is referenced by:  casef1  7088
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