Theorem caseinj 7164

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

Step	Hyp	Ref	Expression
1		df-inl 7122	. . . . . . 7 ⊢ inl = (𝑦 ∈ V ↦ ⟨∅, 𝑦⟩)
2	1	funmpt2 5298	. . . . . 6 ⊢ Fun inl
3		funcnvcnv 5318	. . . . . 6 ⊢ (Fun inl → Fun ◡◡inl)
4	2, 3	ax-mp 5	. . . . 5 ⊢ Fun ◡◡inl
5		caseinj.r	. . . . 5 ⊢ (𝜑 → Fun ◡𝑅)
6		funco 5299	. . . . 5 ⊢ ((Fun ◡◡inl ∧ Fun ◡𝑅) → Fun (◡◡inl ∘ ◡𝑅))
7	4, 5, 6	sylancr 414	. . . 4 ⊢ (𝜑 → Fun (◡◡inl ∘ ◡𝑅))
8		cnvco 4852	. . . . 5 ⊢ ◡(𝑅 ∘ ◡inl) = (◡◡inl ∘ ◡𝑅)
9	8	funeqi 5280	. . . 4 ⊢ (Fun ◡(𝑅 ∘ ◡inl) ↔ Fun (◡◡inl ∘ ◡𝑅))
10	7, 9	sylibr 134	. . 3 ⊢ (𝜑 → Fun ◡(𝑅 ∘ ◡inl))
11		df-inr 7123	. . . . . . 7 ⊢ inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
12	11	funmpt2 5298	. . . . . 6 ⊢ Fun inr
13		funcnvcnv 5318	. . . . . 6 ⊢ (Fun inr → Fun ◡◡inr)
14	12, 13	ax-mp 5	. . . . 5 ⊢ Fun ◡◡inr
15		caseinj.s	. . . . 5 ⊢ (𝜑 → Fun ◡𝑆)
16		funco 5299	. . . . 5 ⊢ ((Fun ◡◡inr ∧ Fun ◡𝑆) → Fun (◡◡inr ∘ ◡𝑆))
17	14, 15, 16	sylancr 414	. . . 4 ⊢ (𝜑 → Fun (◡◡inr ∘ ◡𝑆))
18		cnvco 4852	. . . . 5 ⊢ ◡(𝑆 ∘ ◡inr) = (◡◡inr ∘ ◡𝑆)
19	18	funeqi 5280	. . . 4 ⊢ (Fun ◡(𝑆 ∘ ◡inr) ↔ Fun (◡◡inr ∘ ◡𝑆))
20	17, 19	sylibr 134	. . 3 ⊢ (𝜑 → Fun ◡(𝑆 ∘ ◡inr))
21		df-rn 4675	. . . . . . 7 ⊢ ran (𝑅 ∘ ◡inl) = dom ◡(𝑅 ∘ ◡inl)
22		rncoss 4937	. . . . . . 7 ⊢ ran (𝑅 ∘ ◡inl) ⊆ ran 𝑅
23	21, 22	eqsstrri 3217	. . . . . 6 ⊢ dom ◡(𝑅 ∘ ◡inl) ⊆ ran 𝑅
24		df-rn 4675	. . . . . . 7 ⊢ ran (𝑆 ∘ ◡inr) = dom ◡(𝑆 ∘ ◡inr)
25		rncoss 4937	. . . . . . 7 ⊢ ran (𝑆 ∘ ◡inr) ⊆ ran 𝑆
26	24, 25	eqsstrri 3217	. . . . . 6 ⊢ dom ◡(𝑆 ∘ ◡inr) ⊆ ran 𝑆
27		ss2in 3392	. . . . . 6 ⊢ ((dom ◡(𝑅 ∘ ◡inl) ⊆ ran 𝑅 ∧ dom ◡(𝑆 ∘ ◡inr) ⊆ ran 𝑆) → (dom ◡(𝑅 ∘ ◡inl) ∩ dom ◡(𝑆 ∘ ◡inr)) ⊆ (ran 𝑅 ∩ ran 𝑆))
28	23, 26, 27	mp2an 426	. . . . 5 ⊢ (dom ◡(𝑅 ∘ ◡inl) ∩ dom ◡(𝑆 ∘ ◡inr)) ⊆ (ran 𝑅 ∩ ran 𝑆)
29		caseinj.disj	. . . . 5 ⊢ (𝜑 → (ran 𝑅 ∩ ran 𝑆) = ∅)
30	28, 29	sseqtrid 3234	. . . 4 ⊢ (𝜑 → (dom ◡(𝑅 ∘ ◡inl) ∩ dom ◡(𝑆 ∘ ◡inr)) ⊆ ∅)
31		ss0 3492	. . . 4 ⊢ ((dom ◡(𝑅 ∘ ◡inl) ∩ dom ◡(𝑆 ∘ ◡inr)) ⊆ ∅ → (dom ◡(𝑅 ∘ ◡inl) ∩ dom ◡(𝑆 ∘ ◡inr)) = ∅)
32	30, 31	syl 14	. . 3 ⊢ (𝜑 → (dom ◡(𝑅 ∘ ◡inl) ∩ dom ◡(𝑆 ∘ ◡inr)) = ∅)
33		funun 5303	. . 3 ⊢ (((Fun ◡(𝑅 ∘ ◡inl) ∧ Fun ◡(𝑆 ∘ ◡inr)) ∧ (dom ◡(𝑅 ∘ ◡inl) ∩ dom ◡(𝑆 ∘ ◡inr)) = ∅) → Fun (◡(𝑅 ∘ ◡inl) ∪ ◡(𝑆 ∘ ◡inr)))
34	10, 20, 32, 33	syl21anc 1248	. 2 ⊢ (𝜑 → Fun (◡(𝑅 ∘ ◡inl) ∪ ◡(𝑆 ∘ ◡inr)))
35		df-case 7159	. . . . 5 ⊢ case(𝑅, 𝑆) = ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr))
36	35	cnveqi 4842	. . . 4 ⊢ ◡case(𝑅, 𝑆) = ◡((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr))
37		cnvun 5076	. . . 4 ⊢ ◡((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr)) = (◡(𝑅 ∘ ◡inl) ∪ ◡(𝑆 ∘ ◡inr))
38	36, 37	eqtri 2217	. . 3 ⊢ ◡case(𝑅, 𝑆) = (◡(𝑅 ∘ ◡inl) ∪ ◡(𝑆 ∘ ◡inr))
39	38	funeqi 5280	. 2 ⊢ (Fun ◡case(𝑅, 𝑆) ↔ Fun (◡(𝑅 ∘ ◡inl) ∪ ◡(𝑆 ∘ ◡inr)))
40	34, 39	sylibr 134	1 ⊢ (𝜑 → Fun ◡case(𝑅, 𝑆))

Syntax hints:   → wi 4   = wceq 1364  Vcvv 2763   ∪ cun 3155   ∩ cin 3156   ⊆ wss 3157  ∅c0 3451  ⟨cop 3626  ^◡ccnv 4663  dom cdm 4664  ran crn 4665   ∘ ccom 4668  Fun wfun 5253  1_oc1o 6476  inlcinl 7120  inrcinr 7121  casecdjucase 7158

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-fun 5261  df-inl 7122  df-inr 7123  df-case 7159

This theorem is referenced by:  casef1  7165