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Theorem casef1 7079
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 5413 . . . 4 (𝐹:𝐴1-1𝑋𝐹:𝐴𝑋)
31, 2syl 14 . . 3 (𝜑𝐹:𝐴𝑋)
4 casef1.g . . . 4 (𝜑𝐺:𝐵1-1𝑋)
5 f1f 5413 . . . 4 (𝐺:𝐵1-1𝑋𝐺:𝐵𝑋)
64, 5syl 14 . . 3 (𝜑𝐺:𝐵𝑋)
73, 6casef 7077 . 2 (𝜑 → case(𝐹, 𝐺):(𝐴𝐵)⟶𝑋)
8 df-f1 5213 . . . . 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 5213 . . . . 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 7078 . 2 (𝜑 → Fun case(𝐹, 𝐺))
16 df-f1 5213 . 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 1353  cin 3126  c0 3420  ccnv 4619  ran crn 4621  Fun wfun 5202  wf 5204  1-1wf1 5205  cdju 7026  casecdjucase 7072
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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-v 2737  df-sbc 2961  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-iord 4360  df-on 4362  df-suc 4365  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-1st 6131  df-2nd 6132  df-1o 6407  df-dju 7027  df-inl 7036  df-inr 7037  df-case 7073
This theorem is referenced by:  djudom  7082  exmidsbthrlem  14253
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