Theorem dom2 4411

Description: A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its range. 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 1478	. 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 4410	. 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 146   /\ wa 223   = wceq 958   e. wcel 960   class class class wbr 2624   ~<_ cdom 4371

This theorem is referenced by:  canth2 4490  limenpsi 4511  xpnnen 7500  znnen 7503

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-9 967  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-rep 2698  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-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-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-en 4374  df-dom 4375

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