Theorem caseinj 7279

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 7237	. . . . . . 7 ⊢ inl = (𝑦 ∈ V ↦ ⟨∅, 𝑦⟩)
2	1	funmpt2 5363	. . . . . 6 ⊢ Fun inl
3		funcnvcnv 5386	. . . . . 6 ⊢ (Fun inl → Fun ◡◡inl)
4	2, 3	ax-mp 5	. . . . 5 ⊢ Fun ◡◡inl
5		caseinj.r	. . . . 5 ⊢ (𝜑 → Fun ◡𝑅)
6		funco 5364	. . . . 5 ⊢ ((Fun ◡◡inl ∧ Fun ◡𝑅) → Fun (◡◡inl ∘ ◡𝑅))
7	4, 5, 6	sylancr 414	. . . 4 ⊢ (𝜑 → Fun (◡◡inl ∘ ◡𝑅))
8		cnvco 4913	. . . . 5 ⊢ ◡(𝑅 ∘ ◡inl) = (◡◡inl ∘ ◡𝑅)
9	8	funeqi 5345	. . . 4 ⊢ (Fun ◡(𝑅 ∘ ◡inl) ↔ Fun (◡◡inl ∘ ◡𝑅))
10	7, 9	sylibr 134	. . 3 ⊢ (𝜑 → Fun ◡(𝑅 ∘ ◡inl))
11		df-inr 7238	. . . . . . 7 ⊢ inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
12	11	funmpt2 5363	. . . . . 6 ⊢ Fun inr
13		funcnvcnv 5386	. . . . . 6 ⊢ (Fun inr → Fun ◡◡inr)
14	12, 13	ax-mp 5	. . . . 5 ⊢ Fun ◡◡inr
15		caseinj.s	. . . . 5 ⊢ (𝜑 → Fun ◡𝑆)
16		funco 5364	. . . . 5 ⊢ ((Fun ◡◡inr ∧ Fun ◡𝑆) → Fun (◡◡inr ∘ ◡𝑆))
17	14, 15, 16	sylancr 414	. . . 4 ⊢ (𝜑 → Fun (◡◡inr ∘ ◡𝑆))
18		cnvco 4913	. . . . 5 ⊢ ◡(𝑆 ∘ ◡inr) = (◡◡inr ∘ ◡𝑆)
19	18	funeqi 5345	. . . 4 ⊢ (Fun ◡(𝑆 ∘ ◡inr) ↔ Fun (◡◡inr ∘ ◡𝑆))
20	17, 19	sylibr 134	. . 3 ⊢ (𝜑 → Fun ◡(𝑆 ∘ ◡inr))
21		df-rn 4734	. . . . . . 7 ⊢ ran (𝑅 ∘ ◡inl) = dom ◡(𝑅 ∘ ◡inl)
22		rncoss 5001	. . . . . . 7 ⊢ ran (𝑅 ∘ ◡inl) ⊆ ran 𝑅
23	21, 22	eqsstrri 3258	. . . . . 6 ⊢ dom ◡(𝑅 ∘ ◡inl) ⊆ ran 𝑅
24		df-rn 4734	. . . . . . 7 ⊢ ran (𝑆 ∘ ◡inr) = dom ◡(𝑆 ∘ ◡inr)
25		rncoss 5001	. . . . . . 7 ⊢ ran (𝑆 ∘ ◡inr) ⊆ ran 𝑆
26	24, 25	eqsstrri 3258	. . . . . 6 ⊢ dom ◡(𝑆 ∘ ◡inr) ⊆ ran 𝑆
27		ss2in 3433	. . . . . 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 3275	. . . 4 ⊢ (𝜑 → (dom ◡(𝑅 ∘ ◡inl) ∩ dom ◡(𝑆 ∘ ◡inr)) ⊆ ∅)
31		ss0 3533	. . . 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 5368	. . 3 ⊢ (((Fun ◡(𝑅 ∘ ◡inl) ∧ Fun ◡(𝑆 ∘ ◡inr)) ∧ (dom ◡(𝑅 ∘ ◡inl) ∩ dom ◡(𝑆 ∘ ◡inr)) = ∅) → Fun (◡(𝑅 ∘ ◡inl) ∪ ◡(𝑆 ∘ ◡inr)))
34	10, 20, 32, 33	syl21anc 1270	. 2 ⊢ (𝜑 → Fun (◡(𝑅 ∘ ◡inl) ∪ ◡(𝑆 ∘ ◡inr)))
35		df-case 7274	. . . . 5 ⊢ case(𝑅, 𝑆) = ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr))
36	35	cnveqi 4903	. . . 4 ⊢ ◡case(𝑅, 𝑆) = ◡((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr))
37		cnvun 5140	. . . 4 ⊢ ◡((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr)) = (◡(𝑅 ∘ ◡inl) ∪ ◡(𝑆 ∘ ◡inr))
38	36, 37	eqtri 2250	. . 3 ⊢ ◡case(𝑅, 𝑆) = (◡(𝑅 ∘ ◡inl) ∪ ◡(𝑆 ∘ ◡inr))
39	38	funeqi 5345	. 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 1395  Vcvv 2800   ∪ cun 3196   ∩ cin 3197   ⊆ wss 3198  ∅c0 3492  ⟨cop 3670  ^◡ccnv 4722  dom cdm 4723  ran crn 4724   ∘ ccom 4727  Fun wfun 5318  1_oc1o 6570  inlcinl 7235  inrcinr 7236  casecdjucase 7273

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-fun 5326  df-inl 7237  df-inr 7238  df-case 7274

This theorem is referenced by:  casef1  7280