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Theorem casef1 7119
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 5440 . . . 4 (𝐹:𝐴1-1𝑋𝐹:𝐴𝑋)
31, 2syl 14 . . 3 (𝜑𝐹:𝐴𝑋)
4 casef1.g . . . 4 (𝜑𝐺:𝐵1-1𝑋)
5 f1f 5440 . . . 4 (𝐺:𝐵1-1𝑋𝐺:𝐵𝑋)
64, 5syl 14 . . 3 (𝜑𝐺:𝐵𝑋)
73, 6casef 7117 . 2 (𝜑 → case(𝐹, 𝐺):(𝐴𝐵)⟶𝑋)
8 df-f1 5240 . . . . 5 (𝐹:𝐴1-1𝑋 ↔ (𝐹:𝐴𝑋 ∧ Fun 𝐹))
98simprbi 275 . . . 4 (𝐹:𝐴1-1𝑋 → Fun 𝐹)
101, 9syl 14 . . 3 (𝜑 → Fun 𝐹)
11 df-f1 5240 . . . . 5 (𝐺:𝐵1-1𝑋 ↔ (𝐺:𝐵𝑋 ∧ Fun 𝐺))
1211simprbi 275 . . . 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 7118 . 2 (𝜑 → Fun case(𝐹, 𝐺))
16 df-f1 5240 . 2 (case(𝐹, 𝐺):(𝐴𝐵)–1-1𝑋 ↔ (case(𝐹, 𝐺):(𝐴𝐵)⟶𝑋 ∧ Fun case(𝐹, 𝐺)))
177, 15, 16sylanbrc 417 1 (𝜑 → case(𝐹, 𝐺):(𝐴𝐵)–1-1𝑋)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  cin 3143  c0 3437  ccnv 4643  ran crn 4645  Fun wfun 5229  wf 5231  1-1wf1 5232  cdju 7066  casecdjucase 7112
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-1st 6165  df-2nd 6166  df-1o 6441  df-dju 7067  df-inl 7076  df-inr 7077  df-case 7113
This theorem is referenced by:  djudom  7122  exmidsbthrlem  15229
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