Theorem dom2 4392

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 1473	. 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 4391	. 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 954   e. wcel 956   class class class wbr 2614   ~<_ cdom 4355

This theorem is referenced by:  canth2 4470  limenpsi 4491  xpnnen 7449  znnen 7453

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-9 963  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-rep 2688  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-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-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193  df-en 4357  df-dom 4358

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