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Theorem casef1 6760
Description: The "case" construction of two injective functions with disjoint ranges is an injective function. (Contributed by BJ, 10-Jul-2022.)
Hypotheses
Ref Expression
casef1.f (𝜑𝐹:𝐴1-1𝑋)
casef1.g (𝜑𝐺:𝐵1-1𝑋)
casef1.disj (𝜑 → (ran 𝐹 ∩ ran 𝐺) = ∅)
Assertion
Ref Expression
casef1 (𝜑 → case(𝐹, 𝐺):(𝐴𝐵)–1-1𝑋)

Proof of Theorem casef1
StepHypRef Expression
1 casef1.f . . . 4 (𝜑𝐹:𝐴1-1𝑋)
2 f1f 5200 . . . 4 (𝐹:𝐴1-1𝑋𝐹:𝐴𝑋)
31, 2syl 14 . . 3 (𝜑𝐹:𝐴𝑋)
4 casef1.g . . . 4 (𝜑𝐺:𝐵1-1𝑋)
5 f1f 5200 . . . 4 (𝐺:𝐵1-1𝑋𝐺:𝐵𝑋)
64, 5syl 14 . . 3 (𝜑𝐺:𝐵𝑋)
73, 6casef 6758 . 2 (𝜑 → case(𝐹, 𝐺):(𝐴𝐵)⟶𝑋)
8 df-f1 5007 . . . . 5 (𝐹:𝐴1-1𝑋 ↔ (𝐹:𝐴𝑋 ∧ Fun 𝐹))
98simprbi 269 . . . 4 (𝐹:𝐴1-1𝑋 → Fun 𝐹)
101, 9syl 14 . . 3 (𝜑 → Fun 𝐹)
11 df-f1 5007 . . . . 5 (𝐺:𝐵1-1𝑋 ↔ (𝐺:𝐵𝑋 ∧ Fun 𝐺))
1211simprbi 269 . . . 4 (𝐺:𝐵1-1𝑋 → Fun 𝐺)
134, 12syl 14 . . 3 (𝜑 → Fun 𝐺)
14 casef1.disj . . 3 (𝜑 → (ran 𝐹 ∩ ran 𝐺) = ∅)
1510, 13, 14caseinj 6759 . 2 (𝜑 → Fun case(𝐹, 𝐺))
16 df-f1 5007 . 2 (case(𝐹, 𝐺):(𝐴𝐵)–1-1𝑋 ↔ (case(𝐹, 𝐺):(𝐴𝐵)⟶𝑋 ∧ Fun case(𝐹, 𝐺)))
177, 15, 16sylanbrc 408 1 (𝜑 → case(𝐹, 𝐺):(𝐴𝐵)–1-1𝑋)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  cin 2996  c0 3284  ccnv 4427  ran crn 4429  Fun wfun 4996  wf 4998  1-1wf1 4999  cdju 6709  casecdjucase 6753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-1st 5893  df-2nd 5894  df-1o 6163  df-dju 6710  df-inl 6718  df-inr 6719  df-case 6754
This theorem is referenced by:  djudom  6766  exmidsbthrlem  11569
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