Theorem casef 7086

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 5368	. . . . 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 5368	. . . . 5 ⊢ (𝐺:𝐵⟶𝑋 → Fun 𝐺)
6	4, 5	syl 14	. . . 4 ⊢ (𝜑 → Fun 𝐺)
7	3, 6	casefun 7083	. . 3 ⊢ (𝜑 → Fun case(𝐹, 𝐺))
8		caserel 7085	. . . 4 ⊢ case(𝐹, 𝐺) ⊆ ((dom 𝐹 ⊔ dom 𝐺) × (ran 𝐹 ∪ ran 𝐺))
9		ssid 3175	. . . . 5 ⊢ (dom 𝐹 ⊔ dom 𝐺) ⊆ (dom 𝐹 ⊔ dom 𝐺)
10		frn 5374	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝑋 → ran 𝐹 ⊆ 𝑋)
11	1, 10	syl 14	. . . . . 6 ⊢ (𝜑 → ran 𝐹 ⊆ 𝑋)
12		frn 5374	. . . . . . 7 ⊢ (𝐺:𝐵⟶𝑋 → ran 𝐺 ⊆ 𝑋)
13	4, 12	syl 14	. . . . . 6 ⊢ (𝜑 → ran 𝐺 ⊆ 𝑋)
14	11, 13	unssd 3311	. . . . 5 ⊢ (𝜑 → (ran 𝐹 ∪ ran 𝐺) ⊆ 𝑋)
15		xpss12 4733	. . . . 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 3166	. . 3 ⊢ (𝜑 → case(𝐹, 𝐺) ⊆ ((dom 𝐹 ⊔ dom 𝐺) × 𝑋))
18		funssxp 5385	. . . 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 7084	. . . 4 ⊢ dom case(𝐹, 𝐺) = (dom 𝐹 ⊔ dom 𝐺)
22		fdm 5371	. . . . . 6 ⊢ (𝐹:𝐴⟶𝑋 → dom 𝐹 = 𝐴)
23	1, 22	syl 14	. . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴)
24		fdm 5371	. . . . . 6 ⊢ (𝐺:𝐵⟶𝑋 → dom 𝐺 = 𝐵)
25	4, 24	syl 14	. . . . 5 ⊢ (𝜑 → dom 𝐺 = 𝐵)
26		djueq12 7037	. . . . 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 2222	. . 3 ⊢ (𝜑 → dom case(𝐹, 𝐺) = (𝐴 ⊔ 𝐵))
29	28	feq2d 5353	. 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 1353   ∪ cun 3127   ⊆ wss 3129   × cxp 4624  dom cdm 4626  ran crn 4627  Fun wfun 5210  ⟶wf 5212   ⊔ cdju 7035  casecdjucase 7081

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-suc 4371  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-1st 6140  df-2nd 6141  df-1o 6416  df-dju 7036  df-inl 7045  df-inr 7046  df-case 7082

This theorem is referenced by:  casef1  7088  omp1eomlem  7092  ctm  7107