Theorem fun11uni 3571

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 1709	. . 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 3569	. . 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 1709	. . 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 3570	. . 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 1648   (_ wss 2050  U.cuni 2507  `'ccnv 3175  Fun wfun 3182

This theorem is referenced by:  infxpidmlem7 7559

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-iun 2572  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-fun 3198

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