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Theorem casef 6980
 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 (𝜑𝐹:𝐴𝑋)
casef.g (𝜑𝐺:𝐵𝑋)
Assertion
Ref Expression
casef (𝜑 → case(𝐹, 𝐺):(𝐴𝐵)⟶𝑋)

Proof of Theorem casef
StepHypRef Expression
1 casef.f . . . . 5 (𝜑𝐹:𝐴𝑋)
2 ffun 5282 . . . . 5 (𝐹:𝐴𝑋 → Fun 𝐹)
31, 2syl 14 . . . 4 (𝜑 → Fun 𝐹)
4 casef.g . . . . 5 (𝜑𝐺:𝐵𝑋)
5 ffun 5282 . . . . 5 (𝐺:𝐵𝑋 → Fun 𝐺)
64, 5syl 14 . . . 4 (𝜑 → Fun 𝐺)
73, 6casefun 6977 . . 3 (𝜑 → Fun case(𝐹, 𝐺))
8 caserel 6979 . . . 4 case(𝐹, 𝐺) ⊆ ((dom 𝐹 ⊔ dom 𝐺) × (ran 𝐹 ∪ ran 𝐺))
9 ssid 3121 . . . . 5 (dom 𝐹 ⊔ dom 𝐺) ⊆ (dom 𝐹 ⊔ dom 𝐺)
10 frn 5288 . . . . . . 7 (𝐹:𝐴𝑋 → ran 𝐹𝑋)
111, 10syl 14 . . . . . 6 (𝜑 → ran 𝐹𝑋)
12 frn 5288 . . . . . . 7 (𝐺:𝐵𝑋 → ran 𝐺𝑋)
134, 12syl 14 . . . . . 6 (𝜑 → ran 𝐺𝑋)
1411, 13unssd 3256 . . . . 5 (𝜑 → (ran 𝐹 ∪ ran 𝐺) ⊆ 𝑋)
15 xpss12 4653 . . . . 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))
169, 14, 15sylancr 411 . . . 4 (𝜑 → ((dom 𝐹 ⊔ dom 𝐺) × (ran 𝐹 ∪ ran 𝐺)) ⊆ ((dom 𝐹 ⊔ dom 𝐺) × 𝑋))
178, 16sstrid 3112 . . 3 (𝜑 → case(𝐹, 𝐺) ⊆ ((dom 𝐹 ⊔ dom 𝐺) × 𝑋))
18 funssxp 5299 . . . 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)))
1918simplbi 272 . . 3 ((Fun case(𝐹, 𝐺) ∧ case(𝐹, 𝐺) ⊆ ((dom 𝐹 ⊔ dom 𝐺) × 𝑋)) → case(𝐹, 𝐺):dom case(𝐹, 𝐺)⟶𝑋)
207, 17, 19syl2anc 409 . 2 (𝜑 → case(𝐹, 𝐺):dom case(𝐹, 𝐺)⟶𝑋)
21 casedm 6978 . . . 4 dom case(𝐹, 𝐺) = (dom 𝐹 ⊔ dom 𝐺)
22 fdm 5285 . . . . . 6 (𝐹:𝐴𝑋 → dom 𝐹 = 𝐴)
231, 22syl 14 . . . . 5 (𝜑 → dom 𝐹 = 𝐴)
24 fdm 5285 . . . . . 6 (𝐺:𝐵𝑋 → dom 𝐺 = 𝐵)
254, 24syl 14 . . . . 5 (𝜑 → dom 𝐺 = 𝐵)
26 djueq12 6931 . . . . 5 ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (dom 𝐹 ⊔ dom 𝐺) = (𝐴𝐵))
2723, 25, 26syl2anc 409 . . . 4 (𝜑 → (dom 𝐹 ⊔ dom 𝐺) = (𝐴𝐵))
2821, 27syl5eq 2185 . . 3 (𝜑 → dom case(𝐹, 𝐺) = (𝐴𝐵))
2928feq2d 5267 . 2 (𝜑 → (case(𝐹, 𝐺):dom case(𝐹, 𝐺)⟶𝑋 ↔ case(𝐹, 𝐺):(𝐴𝐵)⟶𝑋))
3020, 29mpbid 146 1 (𝜑 → case(𝐹, 𝐺):(𝐴𝐵)⟶𝑋)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332   ∪ cun 3073   ⊆ wss 3075   × cxp 4544  dom cdm 4546  ran crn 4547  Fun wfun 5124  ⟶wf 5126   ⊔ cdju 6929  casecdjucase 6975 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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2913  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-id 4222  df-iord 4295  df-on 4297  df-suc 4300  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-1st 6045  df-2nd 6046  df-1o 6320  df-dju 6930  df-inl 6939  df-inr 6940  df-case 6976 This theorem is referenced by:  casef1  6982  omp1eomlem  6986  ctm  7001
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