Theorem casef 7379

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 5511	. . . . 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 5511	. . . . 5 ⊢ (𝐺:𝐵⟶𝑋 → Fun 𝐺)
6	4, 5	syl 14	. . . 4 ⊢ (𝜑 → Fun 𝐺)
7	3, 6	casefun 7376	. . 3 ⊢ (𝜑 → Fun case(𝐹, 𝐺))
8		caserel 7378	. . . 4 ⊢ case(𝐹, 𝐺) ⊆ ((dom 𝐹 ⊔ dom 𝐺) × (ran 𝐹 ∪ ran 𝐺))
9		ssid 3258	. . . . 5 ⊢ (dom 𝐹 ⊔ dom 𝐺) ⊆ (dom 𝐹 ⊔ dom 𝐺)
10		frn 5517	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝑋 → ran 𝐹 ⊆ 𝑋)
11	1, 10	syl 14	. . . . . 6 ⊢ (𝜑 → ran 𝐹 ⊆ 𝑋)
12		frn 5517	. . . . . . 7 ⊢ (𝐺:𝐵⟶𝑋 → ran 𝐺 ⊆ 𝑋)
13	4, 12	syl 14	. . . . . 6 ⊢ (𝜑 → ran 𝐺 ⊆ 𝑋)
14	11, 13	unssd 3395	. . . . 5 ⊢ (𝜑 → (ran 𝐹 ∪ ran 𝐺) ⊆ 𝑋)
15		xpss12 4857	. . . . 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 3249	. . 3 ⊢ (𝜑 → case(𝐹, 𝐺) ⊆ ((dom 𝐹 ⊔ dom 𝐺) × 𝑋))
18		funssxp 5532	. . . 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 7377	. . . 4 ⊢ dom case(𝐹, 𝐺) = (dom 𝐹 ⊔ dom 𝐺)
22		fdm 5514	. . . . . 6 ⊢ (𝐹:𝐴⟶𝑋 → dom 𝐹 = 𝐴)
23	1, 22	syl 14	. . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴)
24		fdm 5514	. . . . . 6 ⊢ (𝐺:𝐵⟶𝑋 → dom 𝐺 = 𝐵)
25	4, 24	syl 14	. . . . 5 ⊢ (𝜑 → dom 𝐺 = 𝐵)
26		djueq12 7330	. . . . 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 2277	. . 3 ⊢ (𝜑 → dom case(𝐹, 𝐺) = (𝐴 ⊔ 𝐵))
29	28	feq2d 5496	. 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 1398   ∪ cun 3209   ⊆ wss 3211   × cxp 4747  dom cdm 4749  ran crn 4750  Fun wfun 5346  ⟶wf 5348   ⊔ cdju 7328  casecdjucase 7374

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-suc 4492  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-1st 6334  df-2nd 6335  df-1o 6647  df-dju 7329  df-inl 7338  df-inr 7339  df-case 7375

This theorem is referenced by:  casef1  7381  omp1eomlem  7385  ctm  7400