Theorem casef 7077

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 5360	. . . . 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 5360	. . . . 5 ⊢ (𝐺:𝐵⟶𝑋 → Fun 𝐺)
6	4, 5	syl 14	. . . 4 ⊢ (𝜑 → Fun 𝐺)
7	3, 6	casefun 7074	. . 3 ⊢ (𝜑 → Fun case(𝐹, 𝐺))
8		caserel 7076	. . . 4 ⊢ case(𝐹, 𝐺) ⊆ ((dom 𝐹 ⊔ dom 𝐺) × (ran 𝐹 ∪ ran 𝐺))
9		ssid 3173	. . . . 5 ⊢ (dom 𝐹 ⊔ dom 𝐺) ⊆ (dom 𝐹 ⊔ dom 𝐺)
10		frn 5366	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝑋 → ran 𝐹 ⊆ 𝑋)
11	1, 10	syl 14	. . . . . 6 ⊢ (𝜑 → ran 𝐹 ⊆ 𝑋)
12		frn 5366	. . . . . . 7 ⊢ (𝐺:𝐵⟶𝑋 → ran 𝐺 ⊆ 𝑋)
13	4, 12	syl 14	. . . . . 6 ⊢ (𝜑 → ran 𝐺 ⊆ 𝑋)
14	11, 13	unssd 3309	. . . . 5 ⊢ (𝜑 → (ran 𝐹 ∪ ran 𝐺) ⊆ 𝑋)
15		xpss12 4727	. . . . 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 3164	. . 3 ⊢ (𝜑 → case(𝐹, 𝐺) ⊆ ((dom 𝐹 ⊔ dom 𝐺) × 𝑋))
18		funssxp 5377	. . . 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 7075	. . . 4 ⊢ dom case(𝐹, 𝐺) = (dom 𝐹 ⊔ dom 𝐺)
22		fdm 5363	. . . . . 6 ⊢ (𝐹:𝐴⟶𝑋 → dom 𝐹 = 𝐴)
23	1, 22	syl 14	. . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴)
24		fdm 5363	. . . . . 6 ⊢ (𝐺:𝐵⟶𝑋 → dom 𝐺 = 𝐵)
25	4, 24	syl 14	. . . . 5 ⊢ (𝜑 → dom 𝐺 = 𝐵)
26		djueq12 7028	. . . . 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 2220	. . 3 ⊢ (𝜑 → dom case(𝐹, 𝐺) = (𝐴 ⊔ 𝐵))
29	28	feq2d 5345	. 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 3125   ⊆ wss 3127   × cxp 4618  dom cdm 4620  ran crn 4621  Fun wfun 5202  ⟶wf 5204   ⊔ cdju 7026  casecdjucase 7072

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-v 2737  df-sbc 2961  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-iord 4360  df-on 4362  df-suc 4365  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-1st 6131  df-2nd 6132  df-1o 6407  df-dju 7027  df-inl 7036  df-inr 7037  df-case 7073

This theorem is referenced by:  casef1  7079  omp1eomlem  7083  ctm  7098