Theorem caseinj 7287

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 7245	. . . . . . 7 ⊢ inl = (𝑦 ∈ V ↦ ⟨∅, 𝑦⟩)
2	1	funmpt2 5365	. . . . . 6 ⊢ Fun inl
3		funcnvcnv 5389	. . . . . 6 ⊢ (Fun inl → Fun ◡◡inl)
4	2, 3	ax-mp 5	. . . . 5 ⊢ Fun ◡◡inl
5		caseinj.r	. . . . 5 ⊢ (𝜑 → Fun ◡𝑅)
6		funco 5366	. . . . 5 ⊢ ((Fun ◡◡inl ∧ Fun ◡𝑅) → Fun (◡◡inl ∘ ◡𝑅))
7	4, 5, 6	sylancr 414	. . . 4 ⊢ (𝜑 → Fun (◡◡inl ∘ ◡𝑅))
8		cnvco 4915	. . . . 5 ⊢ ◡(𝑅 ∘ ◡inl) = (◡◡inl ∘ ◡𝑅)
9	8	funeqi 5347	. . . 4 ⊢ (Fun ◡(𝑅 ∘ ◡inl) ↔ Fun (◡◡inl ∘ ◡𝑅))
10	7, 9	sylibr 134	. . 3 ⊢ (𝜑 → Fun ◡(𝑅 ∘ ◡inl))
11		df-inr 7246	. . . . . . 7 ⊢ inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
12	11	funmpt2 5365	. . . . . 6 ⊢ Fun inr
13		funcnvcnv 5389	. . . . . 6 ⊢ (Fun inr → Fun ◡◡inr)
14	12, 13	ax-mp 5	. . . . 5 ⊢ Fun ◡◡inr
15		caseinj.s	. . . . 5 ⊢ (𝜑 → Fun ◡𝑆)
16		funco 5366	. . . . 5 ⊢ ((Fun ◡◡inr ∧ Fun ◡𝑆) → Fun (◡◡inr ∘ ◡𝑆))
17	14, 15, 16	sylancr 414	. . . 4 ⊢ (𝜑 → Fun (◡◡inr ∘ ◡𝑆))
18		cnvco 4915	. . . . 5 ⊢ ◡(𝑆 ∘ ◡inr) = (◡◡inr ∘ ◡𝑆)
19	18	funeqi 5347	. . . 4 ⊢ (Fun ◡(𝑆 ∘ ◡inr) ↔ Fun (◡◡inr ∘ ◡𝑆))
20	17, 19	sylibr 134	. . 3 ⊢ (𝜑 → Fun ◡(𝑆 ∘ ◡inr))
21		df-rn 4736	. . . . . . 7 ⊢ ran (𝑅 ∘ ◡inl) = dom ◡(𝑅 ∘ ◡inl)
22		rncoss 5003	. . . . . . 7 ⊢ ran (𝑅 ∘ ◡inl) ⊆ ran 𝑅
23	21, 22	eqsstrri 3260	. . . . . 6 ⊢ dom ◡(𝑅 ∘ ◡inl) ⊆ ran 𝑅
24		df-rn 4736	. . . . . . 7 ⊢ ran (𝑆 ∘ ◡inr) = dom ◡(𝑆 ∘ ◡inr)
25		rncoss 5003	. . . . . . 7 ⊢ ran (𝑆 ∘ ◡inr) ⊆ ran 𝑆
26	24, 25	eqsstrri 3260	. . . . . 6 ⊢ dom ◡(𝑆 ∘ ◡inr) ⊆ ran 𝑆
27		ss2in 3435	. . . . . 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 3277	. . . 4 ⊢ (𝜑 → (dom ◡(𝑅 ∘ ◡inl) ∩ dom ◡(𝑆 ∘ ◡inr)) ⊆ ∅)
31		ss0 3535	. . . 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 5371	. . 3 ⊢ (((Fun ◡(𝑅 ∘ ◡inl) ∧ Fun ◡(𝑆 ∘ ◡inr)) ∧ (dom ◡(𝑅 ∘ ◡inl) ∩ dom ◡(𝑆 ∘ ◡inr)) = ∅) → Fun (◡(𝑅 ∘ ◡inl) ∪ ◡(𝑆 ∘ ◡inr)))
34	10, 20, 32, 33	syl21anc 1272	. 2 ⊢ (𝜑 → Fun (◡(𝑅 ∘ ◡inl) ∪ ◡(𝑆 ∘ ◡inr)))
35		df-case 7282	. . . . 5 ⊢ case(𝑅, 𝑆) = ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr))
36	35	cnveqi 4905	. . . 4 ⊢ ◡case(𝑅, 𝑆) = ◡((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr))
37		cnvun 5142	. . . 4 ⊢ ◡((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr)) = (◡(𝑅 ∘ ◡inl) ∪ ◡(𝑆 ∘ ◡inr))
38	36, 37	eqtri 2252	. . 3 ⊢ ◡case(𝑅, 𝑆) = (◡(𝑅 ∘ ◡inl) ∪ ◡(𝑆 ∘ ◡inr))
39	38	funeqi 5347	. 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 1397  Vcvv 2802   ∪ cun 3198   ∩ cin 3199   ⊆ wss 3200  ∅c0 3494  ⟨cop 3672  ^◡ccnv 4724  dom cdm 4725  ran crn 4726   ∘ ccom 4729  Fun wfun 5320  1_oc1o 6574  inlcinl 7243  inrcinr 7244  casecdjucase 7281

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-fun 5328  df-inl 7245  df-inr 7246  df-case 7282

This theorem is referenced by:  casef1  7288