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Theorem casef1 6941
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 5296 . . . 4 (𝐹:𝐴1-1𝑋𝐹:𝐴𝑋)
31, 2syl 14 . . 3 (𝜑𝐹:𝐴𝑋)
4 casef1.g . . . 4 (𝜑𝐺:𝐵1-1𝑋)
5 f1f 5296 . . . 4 (𝐺:𝐵1-1𝑋𝐺:𝐵𝑋)
64, 5syl 14 . . 3 (𝜑𝐺:𝐵𝑋)
73, 6casef 6939 . 2 (𝜑 → case(𝐹, 𝐺):(𝐴𝐵)⟶𝑋)
8 df-f1 5096 . . . . 5 (𝐹:𝐴1-1𝑋 ↔ (𝐹:𝐴𝑋 ∧ Fun 𝐹))
98simprbi 271 . . . 4 (𝐹:𝐴1-1𝑋 → Fun 𝐹)
101, 9syl 14 . . 3 (𝜑 → Fun 𝐹)
11 df-f1 5096 . . . . 5 (𝐺:𝐵1-1𝑋 ↔ (𝐺:𝐵𝑋 ∧ Fun 𝐺))
1211simprbi 271 . . . 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 6940 . 2 (𝜑 → Fun case(𝐹, 𝐺))
16 df-f1 5096 . 2 (case(𝐹, 𝐺):(𝐴𝐵)–1-1𝑋 ↔ (case(𝐹, 𝐺):(𝐴𝐵)⟶𝑋 ∧ Fun case(𝐹, 𝐺)))
177, 15, 16sylanbrc 411 1 (𝜑 → case(𝐹, 𝐺):(𝐴𝐵)–1-1𝑋)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1314  cin 3038  c0 3331  ccnv 4506  ran crn 4508  Fun wfun 5085  wf 5087  1-1wf1 5088  cdju 6888  casecdjucase 6934
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-suc 4261  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-1st 6004  df-2nd 6005  df-1o 6279  df-dju 6889  df-inl 6898  df-inr 6899  df-case 6935
This theorem is referenced by:  djudom  6944  exmidsbthrlem  13028
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