Theorem dom2 5464

Description: A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. C and D can be read C(x) and D(y), as can be shown from their distinct variable conditions.

Hypotheses

Ref	Expression
dom2.1	|- (x e. A -> C e. B)
dom2.2	|- ((x e. A /\ y e. A) -> (C = D <-> x = y))

Assertion

Ref	Expression
dom2	|- (A e. R -> A ~<_ B)

Distinct variable groups:   x,y,A   x,B,y   y,C   x,D

Proof of Theorem dom2

Step	Hyp	Ref	Expression
1		eqid 1884	. 2 |- A = A
2		dom2.1	. . . 4 |- (x e. A -> C e. B)
3	2	a1i 8	. . 3 |- (A = A -> (x e. A -> C e. B))
4		dom2.2	. . . 4 |- ((x e. A /\ y e. A) -> (C = D <-> x = y))
5	4	a1i 8	. . 3 |- (A = A -> ((x e. A /\ y e. A) -> (C = D <-> x = y)))
6	3, 5	dom2d 5463	. 2 |- (A = A -> (A e. R -> A ~<_ B))
7	1, 6	ax-mp 7	1 |- (A e. R -> A ~<_ B)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   class class class wbr 3338   ~<_ cdom 5424

This theorem is referenced by:  canth2 5548  limenpsi 5599  xpnnen 8768  znnen 8771

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-en 5427  df-dom 5428

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