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Theorem casef 7087
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 5369 . . . . 5 (𝐹:𝐴𝑋 → Fun 𝐹)
31, 2syl 14 . . . 4 (𝜑 → Fun 𝐹)
4 casef.g . . . . 5 (𝜑𝐺:𝐵𝑋)
5 ffun 5369 . . . . 5 (𝐺:𝐵𝑋 → Fun 𝐺)
64, 5syl 14 . . . 4 (𝜑 → Fun 𝐺)
73, 6casefun 7084 . . 3 (𝜑 → Fun case(𝐹, 𝐺))
8 caserel 7086 . . . 4 case(𝐹, 𝐺) ⊆ ((dom 𝐹 ⊔ dom 𝐺) × (ran 𝐹 ∪ ran 𝐺))
9 ssid 3176 . . . . 5 (dom 𝐹 ⊔ dom 𝐺) ⊆ (dom 𝐹 ⊔ dom 𝐺)
10 frn 5375 . . . . . . 7 (𝐹:𝐴𝑋 → ran 𝐹𝑋)
111, 10syl 14 . . . . . 6 (𝜑 → ran 𝐹𝑋)
12 frn 5375 . . . . . . 7 (𝐺:𝐵𝑋 → ran 𝐺𝑋)
134, 12syl 14 . . . . . 6 (𝜑 → ran 𝐺𝑋)
1411, 13unssd 3312 . . . . 5 (𝜑 → (ran 𝐹 ∪ ran 𝐺) ⊆ 𝑋)
15 xpss12 4734 . . . . 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 414 . . . 4 (𝜑 → ((dom 𝐹 ⊔ dom 𝐺) × (ran 𝐹 ∪ ran 𝐺)) ⊆ ((dom 𝐹 ⊔ dom 𝐺) × 𝑋))
178, 16sstrid 3167 . . 3 (𝜑 → case(𝐹, 𝐺) ⊆ ((dom 𝐹 ⊔ dom 𝐺) × 𝑋))
18 funssxp 5386 . . . 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 274 . . 3 ((Fun case(𝐹, 𝐺) ∧ case(𝐹, 𝐺) ⊆ ((dom 𝐹 ⊔ dom 𝐺) × 𝑋)) → case(𝐹, 𝐺):dom case(𝐹, 𝐺)⟶𝑋)
207, 17, 19syl2anc 411 . 2 (𝜑 → case(𝐹, 𝐺):dom case(𝐹, 𝐺)⟶𝑋)
21 casedm 7085 . . . 4 dom case(𝐹, 𝐺) = (dom 𝐹 ⊔ dom 𝐺)
22 fdm 5372 . . . . . 6 (𝐹:𝐴𝑋 → dom 𝐹 = 𝐴)
231, 22syl 14 . . . . 5 (𝜑 → dom 𝐹 = 𝐴)
24 fdm 5372 . . . . . 6 (𝐺:𝐵𝑋 → dom 𝐺 = 𝐵)
254, 24syl 14 . . . . 5 (𝜑 → dom 𝐺 = 𝐵)
26 djueq12 7038 . . . . 5 ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (dom 𝐹 ⊔ dom 𝐺) = (𝐴𝐵))
2723, 25, 26syl2anc 411 . . . 4 (𝜑 → (dom 𝐹 ⊔ dom 𝐺) = (𝐴𝐵))
2821, 27eqtrid 2222 . . 3 (𝜑 → dom case(𝐹, 𝐺) = (𝐴𝐵))
2928feq2d 5354 . 2 (𝜑 → (case(𝐹, 𝐺):dom case(𝐹, 𝐺)⟶𝑋 ↔ case(𝐹, 𝐺):(𝐴𝐵)⟶𝑋))
3020, 29mpbid 147 1 (𝜑 → case(𝐹, 𝐺):(𝐴𝐵)⟶𝑋)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  cun 3128  wss 3130   × cxp 4625  dom cdm 4627  ran crn 4628  Fun wfun 5211  wf 5213  cdju 7036  casecdjucase 7082
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 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434
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 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-1st 6141  df-2nd 6142  df-1o 6417  df-dju 7037  df-inl 7046  df-inr 7047  df-case 7083
This theorem is referenced by:  casef1  7089  omp1eomlem  7093  ctm  7108
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