Theorem casef 7163

Description: The "case" construction of two functions is a function on the disjoint union of their domains. (Contributed by BJ, 10-Jul-2022.)

Hypotheses

Ref	Expression
casef.f	⊢ (𝜑 → 𝐹:𝐴⟶𝑋)
casef.g	⊢ (𝜑 → 𝐺:𝐵⟶𝑋)

Assertion

Ref	Expression
casef	⊢ (𝜑 → case(𝐹, 𝐺):(𝐴 ⊔ 𝐵)⟶𝑋)

Proof of Theorem casef

Step	Hyp	Ref	Expression
1		casef.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝑋)
2		ffun 5413	. . . . 5 ⊢ (𝐹:𝐴⟶𝑋 → Fun 𝐹)
3	1, 2	syl 14	. . . 4 ⊢ (𝜑 → Fun 𝐹)
4		casef.g	. . . . 5 ⊢ (𝜑 → 𝐺:𝐵⟶𝑋)
5		ffun 5413	. . . . 5 ⊢ (𝐺:𝐵⟶𝑋 → Fun 𝐺)
6	4, 5	syl 14	. . . 4 ⊢ (𝜑 → Fun 𝐺)
7	3, 6	casefun 7160	. . 3 ⊢ (𝜑 → Fun case(𝐹, 𝐺))
8		caserel 7162	. . . 4 ⊢ case(𝐹, 𝐺) ⊆ ((dom 𝐹 ⊔ dom 𝐺) × (ran 𝐹 ∪ ran 𝐺))
9		ssid 3204	. . . . 5 ⊢ (dom 𝐹 ⊔ dom 𝐺) ⊆ (dom 𝐹 ⊔ dom 𝐺)
10		frn 5419	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝑋 → ran 𝐹 ⊆ 𝑋)
11	1, 10	syl 14	. . . . . 6 ⊢ (𝜑 → ran 𝐹 ⊆ 𝑋)
12		frn 5419	. . . . . . 7 ⊢ (𝐺:𝐵⟶𝑋 → ran 𝐺 ⊆ 𝑋)
13	4, 12	syl 14	. . . . . 6 ⊢ (𝜑 → ran 𝐺 ⊆ 𝑋)
14	11, 13	unssd 3340	. . . . 5 ⊢ (𝜑 → (ran 𝐹 ∪ ran 𝐺) ⊆ 𝑋)
15		xpss12 4771	. . . . 5 ⊢ (((dom 𝐹 ⊔ dom 𝐺) ⊆ (dom 𝐹 ⊔ dom 𝐺) ∧ (ran 𝐹 ∪ ran 𝐺) ⊆ 𝑋) → ((dom 𝐹 ⊔ dom 𝐺) × (ran 𝐹 ∪ ran 𝐺)) ⊆ ((dom 𝐹 ⊔ dom 𝐺) × 𝑋))
16	9, 14, 15	sylancr 414	. . . 4 ⊢ (𝜑 → ((dom 𝐹 ⊔ dom 𝐺) × (ran 𝐹 ∪ ran 𝐺)) ⊆ ((dom 𝐹 ⊔ dom 𝐺) × 𝑋))
17	8, 16	sstrid 3195	. . 3 ⊢ (𝜑 → case(𝐹, 𝐺) ⊆ ((dom 𝐹 ⊔ dom 𝐺) × 𝑋))
18		funssxp 5430	. . . 4 ⊢ ((Fun case(𝐹, 𝐺) ∧ case(𝐹, 𝐺) ⊆ ((dom 𝐹 ⊔ dom 𝐺) × 𝑋)) ↔ (case(𝐹, 𝐺):dom case(𝐹, 𝐺)⟶𝑋 ∧ dom case(𝐹, 𝐺) ⊆ (dom 𝐹 ⊔ dom 𝐺)))
19	18	simplbi 274	. . 3 ⊢ ((Fun case(𝐹, 𝐺) ∧ case(𝐹, 𝐺) ⊆ ((dom 𝐹 ⊔ dom 𝐺) × 𝑋)) → case(𝐹, 𝐺):dom case(𝐹, 𝐺)⟶𝑋)
20	7, 17, 19	syl2anc 411	. 2 ⊢ (𝜑 → case(𝐹, 𝐺):dom case(𝐹, 𝐺)⟶𝑋)
21		casedm 7161	. . . 4 ⊢ dom case(𝐹, 𝐺) = (dom 𝐹 ⊔ dom 𝐺)
22		fdm 5416	. . . . . 6 ⊢ (𝐹:𝐴⟶𝑋 → dom 𝐹 = 𝐴)
23	1, 22	syl 14	. . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴)
24		fdm 5416	. . . . . 6 ⊢ (𝐺:𝐵⟶𝑋 → dom 𝐺 = 𝐵)
25	4, 24	syl 14	. . . . 5 ⊢ (𝜑 → dom 𝐺 = 𝐵)
26		djueq12 7114	. . . . 5 ⊢ ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (dom 𝐹 ⊔ dom 𝐺) = (𝐴 ⊔ 𝐵))
27	23, 25, 26	syl2anc 411	. . . 4 ⊢ (𝜑 → (dom 𝐹 ⊔ dom 𝐺) = (𝐴 ⊔ 𝐵))
28	21, 27	eqtrid 2241	. . 3 ⊢ (𝜑 → dom case(𝐹, 𝐺) = (𝐴 ⊔ 𝐵))
29	28	feq2d 5398	. 2 ⊢ (𝜑 → (case(𝐹, 𝐺):dom case(𝐹, 𝐺)⟶𝑋 ↔ case(𝐹, 𝐺):(𝐴 ⊔ 𝐵)⟶𝑋))
30	20, 29	mpbid 147	1 ⊢ (𝜑 → case(𝐹, 𝐺):(𝐴 ⊔ 𝐵)⟶𝑋)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∪ cun 3155   ⊆ wss 3157   × cxp 4662  dom cdm 4664  ran crn 4665  Fun wfun 5253  ⟶wf 5255   ⊔ cdju 7112  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-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  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-ne 2368  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  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-uni 3841  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-suc 4407  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-1st 6207  df-2nd 6208  df-1o 6483  df-dju 7113  df-inl 7122  df-inr 7123  df-case 7159

This theorem is referenced by:  casef1  7165  omp1eomlem  7169  ctm  7184