Theorem casef 7053

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 5340	. . . . 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 5340	. . . . 5 ⊢ (𝐺:𝐵⟶𝑋 → Fun 𝐺)
6	4, 5	syl 14	. . . 4 ⊢ (𝜑 → Fun 𝐺)
7	3, 6	casefun 7050	. . 3 ⊢ (𝜑 → Fun case(𝐹, 𝐺))
8		caserel 7052	. . . 4 ⊢ case(𝐹, 𝐺) ⊆ ((dom 𝐹 ⊔ dom 𝐺) × (ran 𝐹 ∪ ran 𝐺))
9		ssid 3162	. . . . 5 ⊢ (dom 𝐹 ⊔ dom 𝐺) ⊆ (dom 𝐹 ⊔ dom 𝐺)
10		frn 5346	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝑋 → ran 𝐹 ⊆ 𝑋)
11	1, 10	syl 14	. . . . . 6 ⊢ (𝜑 → ran 𝐹 ⊆ 𝑋)
12		frn 5346	. . . . . . 7 ⊢ (𝐺:𝐵⟶𝑋 → ran 𝐺 ⊆ 𝑋)
13	4, 12	syl 14	. . . . . 6 ⊢ (𝜑 → ran 𝐺 ⊆ 𝑋)
14	11, 13	unssd 3298	. . . . 5 ⊢ (𝜑 → (ran 𝐹 ∪ ran 𝐺) ⊆ 𝑋)
15		xpss12 4711	. . . . 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 411	. . . 4 ⊢ (𝜑 → ((dom 𝐹 ⊔ dom 𝐺) × (ran 𝐹 ∪ ran 𝐺)) ⊆ ((dom 𝐹 ⊔ dom 𝐺) × 𝑋))
17	8, 16	sstrid 3153	. . 3 ⊢ (𝜑 → case(𝐹, 𝐺) ⊆ ((dom 𝐹 ⊔ dom 𝐺) × 𝑋))
18		funssxp 5357	. . . 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 272	. . 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 409	. 2 ⊢ (𝜑 → case(𝐹, 𝐺):dom case(𝐹, 𝐺)⟶𝑋)
21		casedm 7051	. . . 4 ⊢ dom case(𝐹, 𝐺) = (dom 𝐹 ⊔ dom 𝐺)
22		fdm 5343	. . . . . 6 ⊢ (𝐹:𝐴⟶𝑋 → dom 𝐹 = 𝐴)
23	1, 22	syl 14	. . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴)
24		fdm 5343	. . . . . 6 ⊢ (𝐺:𝐵⟶𝑋 → dom 𝐺 = 𝐵)
25	4, 24	syl 14	. . . . 5 ⊢ (𝜑 → dom 𝐺 = 𝐵)
26		djueq12 7004	. . . . 5 ⊢ ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (dom 𝐹 ⊔ dom 𝐺) = (𝐴 ⊔ 𝐵))
27	23, 25, 26	syl2anc 409	. . . 4 ⊢ (𝜑 → (dom 𝐹 ⊔ dom 𝐺) = (𝐴 ⊔ 𝐵))
28	21, 27	syl5eq 2211	. . 3 ⊢ (𝜑 → dom case(𝐹, 𝐺) = (𝐴 ⊔ 𝐵))
29	28	feq2d 5325	. 2 ⊢ (𝜑 → (case(𝐹, 𝐺):dom case(𝐹, 𝐺)⟶𝑋 ↔ case(𝐹, 𝐺):(𝐴 ⊔ 𝐵)⟶𝑋))
30	20, 29	mpbid 146	1 ⊢ (𝜑 → case(𝐹, 𝐺):(𝐴 ⊔ 𝐵)⟶𝑋)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1343   ∪ cun 3114   ⊆ wss 3116   × cxp 4602  dom cdm 4604  ran crn 4605  Fun wfun 5182  ⟶wf 5184   ⊔ cdju 7002  casecdjucase 7048

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-1st 6108  df-2nd 6109  df-1o 6384  df-dju 7003  df-inl 7012  df-inr 7013  df-case 7049

This theorem is referenced by:  casef1  7055  omp1eomlem  7059  ctm  7074