Theorem fun11uni 3565

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		pm3.26 319	. . . . 5 |- ((Fun f /\ Fun `'f) -> Fun f)
2	1	anim1i 334	. . . 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 1706	. . 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 3563	. . 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		pm3.27 323	. . . . 5 |- ((Fun f /\ Fun `'f) -> Fun `'f)
7	6	anim1i 334	. . . 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 1706	. . 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 3564	. . 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 288	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 222   /\ wa 223  A.wral 1645   (_ wss 2047  U.cuni 2503  `'ccnv 3169  Fun wfun 3176

This theorem is referenced by:  infxpidmlem7 7558

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-iun 2568  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-fun 3192

Copyright terms: Public domain