Theorem fun11uni 4483

Description: The union of a chain (with respect to inclusion) of one-to-one functions is a one-to-one function.

Assertion

Ref	Expression
fun11uni	|- (A.f e. A ((Fun f /\ Fun `'f) /\ A.g e. A (f C_ g \/ g C_ f)) -> (Fun U.A /\ Fun `'U.A))

Distinct variable group:   f,g,A

Proof of Theorem fun11uni

Step	Hyp	Ref	Expression
1		simpl 346	. . . . 5 |- ((Fun f /\ Fun `'f) -> Fun f)
2	1	anim1i 361	. . . 4 |- (((Fun f /\ Fun `'f) /\ A.g e. A (f C_ g \/ g C_ f)) -> (Fun f /\ A.g e. A (f C_ g \/ g C_ f)))
3	2	ralimi 2168	. . 3 |- (A.f e. A ((Fun f /\ Fun `'f) /\ A.g e. A (f C_ g \/ g C_ f)) -> A.f e. A (Fun f /\ A.g e. A (f C_ g \/ g C_ f)))
4		fununi 4481	. . 3 |- (A.f e. A (Fun f /\ A.g e. A (f C_ g \/ g C_ f)) -> Fun U.A)
5	3, 4	syl 12	. 2 |- (A.f e. A ((Fun f /\ Fun `'f) /\ A.g e. A (f C_ g \/ g C_ f)) -> Fun U.A)
6		simpr 350	. . . . 5 |- ((Fun f /\ Fun `'f) -> Fun `'f)
7	6	anim1i 361	. . . 4 |- (((Fun f /\ Fun `'f) /\ A.g e. A (f C_ g \/ g C_ f)) -> (Fun `'f /\ A.g e. A (f C_ g \/ g C_ f)))
8	7	ralimi 2168	. . 3 |- (A.f e. A ((Fun f /\ Fun `'f) /\ A.g e. A (f C_ g \/ g C_ f)) -> A.f e. A (Fun `'f /\ A.g e. A (f C_ g \/ g C_ f)))
9		funcnvuni 4482	. . 3 |- (A.f e. A (Fun `'f /\ A.g e. A (f C_ g \/ g C_ f)) -> Fun `'U.A)
10	8, 9	syl 12	. 2 |- (A.f e. A ((Fun f /\ Fun `'f) /\ A.g e. A (f C_ g \/ g C_ f)) -> Fun `'U.A)
11	5, 10	jca 310	1 |- (A.f e. A ((Fun f /\ Fun `'f) /\ A.g e. A (f C_ g \/ g C_ f)) -> (Fun U.A /\ Fun `'U.A))

Syntax hints:   -> wi 3   \/ wo 239   /\ wa 240  A.wral 2105   C_ wss 2593  U.cuni 3177  `'ccnv 3985  Fun wfun 3992

This theorem is referenced by:  infxpidmlem7 8827

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-fun 4008

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