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Theorem fun11uni 7948
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 (∀𝑓𝐴 ((Fun 𝑓 ∧ Fun 𝑓) ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → (Fun 𝐴 ∧ Fun 𝐴))
Distinct variable group:   𝑓,𝑔,𝐴

Proof of Theorem fun11uni
StepHypRef Expression
1 simpl 481 . . . . 5 ((Fun 𝑓 ∧ Fun 𝑓) → Fun 𝑓)
21anim1i 613 . . . 4 (((Fun 𝑓 ∧ Fun 𝑓) ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)))
32ralimi 3080 . . 3 (∀𝑓𝐴 ((Fun 𝑓 ∧ Fun 𝑓) ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → ∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)))
4 fununi 6633 . . 3 (∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → Fun 𝐴)
53, 4syl 17 . 2 (∀𝑓𝐴 ((Fun 𝑓 ∧ Fun 𝑓) ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → Fun 𝐴)
6 simpr 483 . . . . 5 ((Fun 𝑓 ∧ Fun 𝑓) → Fun 𝑓)
76anim1i 613 . . . 4 (((Fun 𝑓 ∧ Fun 𝑓) ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)))
87ralimi 3080 . . 3 (∀𝑓𝐴 ((Fun 𝑓 ∧ Fun 𝑓) ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → ∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)))
9 funcnvuni 7947 . . 3 (∀𝑓𝐴 (Fun 𝑓 ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → Fun 𝐴)
108, 9syl 17 . 2 (∀𝑓𝐴 ((Fun 𝑓 ∧ Fun 𝑓) ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → Fun 𝐴)
115, 10jca 510 1 (∀𝑓𝐴 ((Fun 𝑓 ∧ Fun 𝑓) ∧ ∀𝑔𝐴 (𝑓𝑔𝑔𝑓)) → (Fun 𝐴 ∧ Fun 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wo 845  wral 3058  wss 3949   cuni 4912  ccnv 5681  Fun wfun 6547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-fun 6555
This theorem is referenced by:  f1iun  7955
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