Theorem fun11uni 3640

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 (_ g \/ g (_ f)) -> (Fun U.A /\ Fun `'U.A))

Distinct variable group:   f,g,A

Proof of Theorem fun11uni

Step	Hyp	Ref	Expression
1		pm3.26 317	. . . . 5 |- ((Fun f /\ Fun `'f) -> Fun f)
2	1	anim1i 332	. . . 4 |- (((Fun f /\ Fun `'f) /\ A.g e. A (f (_ g \/ g (_ f)) -> (Fun f /\ A.g e. A (f (_ g \/ g (_ f)))
3	2	r19.20si 1744	. . 3 |- (A.f e. A ((Fun f /\ Fun `'f) /\ A.g e. A (f (_ g \/ g (_ f)) -> A.f e. A (Fun f /\ A.g e. A (f (_ g \/ g (_ f)))
4		fununi 3638	. . 3 |- (A.f e. A (Fun f /\ A.g e. A (f (_ g \/ g (_ f)) -> Fun U.A)
5	3, 4	syl 10	. 2 |- (A.f e. A ((Fun f /\ Fun `'f) /\ A.g e. A (f (_ g \/ g (_ f)) -> Fun U.A)
6		pm3.27 321	. . . . 5 |- ((Fun f /\ Fun `'f) -> Fun `'f)
7	6	anim1i 332	. . . 4 |- (((Fun f /\ Fun `'f) /\ A.g e. A (f (_ g \/ g (_ f)) -> (Fun `'f /\ A.g e. A (f (_ g \/ g (_ f)))
8	7	r19.20si 1744	. . 3 |- (A.f e. A ((Fun f /\ Fun `'f) /\ A.g e. A (f (_ g \/ g (_ f)) -> A.f e. A (Fun `'f /\ A.g e. A (f (_ g \/ g (_ f)))
9		funcnvuni 3639	. . 3 |- (A.f e. A (Fun `'f /\ A.g e. A (f (_ g \/ g (_ f)) -> Fun `'U.A)
10	8, 9	syl 10	. 2 |- (A.f e. A ((Fun f /\ Fun `'f) /\ A.g e. A (f (_ g \/ g (_ f)) -> Fun `'U.A)
11	5, 10	jca 286	1 |- (A.f e. A ((Fun f /\ Fun `'f) /\ A.g e. A (f (_ g \/ g (_ f)) -> (Fun U.A /\ Fun `'U.A))

Syntax hints:   -> wi 3   \/ wo 220   /\ wa 221  A.wral 1683   (_ wss 2091  U.cuni 2551  `'ccnv 3224  Fun wfun 3231

This theorem is referenced by:  infxpidmlem7 7683

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 994  ax-gen 995  ax-8 996  ax-10 998  ax-11 999  ax-12 1000  ax-13 1001  ax-14 1002  ax-17 1003  ax-4 1005  ax-5o 1007  ax-6o 1010  ax-9o 1155  ax-10o 1173  ax-16 1243  ax-11o 1251  ax-ext 1494  ax-sep 2754  ax-pow 2794  ax-pr 2832  ax-un 2920

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1013  df-sb 1205  df-eu 1415  df-mo 1416  df-clab 1500  df-cleq 1505  df-clel 1508  df-ne 1624  df-ral 1687  df-rex 1688  df-v 1850  df-dif 2093  df-un 2094  df-in 2095  df-ss 2097  df-nul 2325  df-pw 2447  df-sn 2457  df-pr 2458  df-op 2461  df-uni 2552  df-iun 2616  df-br 2670  df-opab 2718  df-id 2889  df-xp 3239  df-rel 3240  df-cnv 3241  df-co 3242  df-dm 3243  df-rn 3244  df-fun 3247

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