Theorem fun11uni 3557

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 1703	. . 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 3555	. . 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 1703	. . 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 3556	. . 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 1642   (_ wss 2043  U.cuni 2498  `'ccnv 3164  Fun wfun 3171

This theorem is referenced by:  infxpidmlem7 7509

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-iun 2563  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-fun 3187

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