Theorem f1oen2g 4375

Description: The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 4376 does not require the Axiom of Replacement.

Assertion

Ref	Expression
f1oen2g	|- ((F e. C /\ F:A-1-1-onto->B) -> A ~~ B)

Proof of Theorem f1oen2g

Step	Hyp	Ref	Expression
1		brprc 2651	. . . . . 6 |- (-. B e. V -> (A ~~ B <-> A ~~ A))
2		enrefg 4371	. . . . . 6 |- (A e. V -> A ~~ A)
3	1, 2	syl5bir 210	. . . . 5 |- (-. B e. V -> (A e. V -> A ~~ B))
4	3	a1d 12	. . . 4 |- (-. B e. V -> (E.f f:A-1-1-onto->B -> (A e. V -> A ~~ B)))
5	4	com3r 35	. . 3 |- (A e. V -> (-. B e. V -> (E.f f:A-1-1-onto->B -> A ~~ B)))
6		breng 4357	. . . 4 |- (B e. V -> (A ~~ B <-> E.f f:A-1-1-onto->B))
7	6	biimprd 154	. . 3 |- (B e. V -> (E.f f:A-1-1-onto->B -> A ~~ B))
8	5, 7	pm2.61d2 129	. 2 |- (A e. V -> (E.f f:A-1-1-onto->B -> A ~~ B))
9		dmfex 3640	. . 3 |- ((F e. C /\ F:A-->B) -> A e. V)
10		f1of 3674	. . 3 |- (F:A-1-1-onto->B -> F:A-->B)
11	9, 10	sylan2 451	. 2 |- ((F e. C /\ F:A-1-1-onto->B) -> A e. V)
12		f1oeq1 3669	. . . 4 |- (f = F -> (f:A-1-1-onto->B <-> F:A-1-1-onto->B))
13	12	cla4egv 1854	. . 3 |- (F e. C -> (F:A-1-1-onto->B -> E.f f:A-1-1-onto->B))
14	13	imp 350	. 2 |- ((F e. C /\ F:A-1-1-onto->B) -> E.f f:A-1-1-onto->B)
15	8, 11, 14	sylc 68	1 |- ((F e. C /\ F:A-1-1-onto->B) -> A ~~ B)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   e. wcel 955  E.wex 977  Vcvv 1802   class class class wbr 2609  -->wf 3168  -1-1-onto->wf1o 3171   ~~ cen 4348

This theorem is referenced by:  f1oeng 4376

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-en 4351

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