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Theorem fun11uni 4912
Description: The union of a chain (with respect to inclusion) of one-to-one functions is a one-to-one function. (Contributed by NM, 11-Aug-2004.)
Assertion
Ref Expression
fun11uni (f A ((Fun f Fun f) g A (fg gf)) → (Fun A Fun A))
Distinct variable group:   f,g,A

Proof of Theorem fun11uni
StepHypRef Expression
1 simpl 102 . . . . 5 ((Fun f Fun f) → Fun f)
21anim1i 323 . . . 4 (((Fun f Fun f) g A (fg gf)) → (Fun f g A (fg gf)))
32ralimi 2378 . . 3 (f A ((Fun f Fun f) g A (fg gf)) → f A (Fun f g A (fg gf)))
4 fununi 4910 . . 3 (f A (Fun f g A (fg gf)) → Fun A)
53, 4syl 14 . 2 (f A ((Fun f Fun f) g A (fg gf)) → Fun A)
6 simpr 103 . . . . 5 ((Fun f Fun f) → Fun f)
76anim1i 323 . . . 4 (((Fun f Fun f) g A (fg gf)) → (Fun f g A (fg gf)))
87ralimi 2378 . . 3 (f A ((Fun f Fun f) g A (fg gf)) → f A (Fun f g A (fg gf)))
9 funcnvuni 4911 . . 3 (f A (Fun f g A (fg gf)) → Fun A)
108, 9syl 14 . 2 (f A ((Fun f Fun f) g A (fg gf)) → Fun A)
115, 10jca 290 1 (f A ((Fun f Fun f) g A (fg gf)) → (Fun A Fun A))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wo 628  wral 2300  wss 2911   cuni 3571  ccnv 4287  Fun wfun 4839
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847
This theorem is referenced by:  fun11iun  5090
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