Theorem dom2 4546

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 1518	. 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 4545	. 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 144   /\ wa 221   = wceq 992   e. wcel 994   class class class wbr 2692   ~<_ cdom 4506

This theorem is referenced by:  canth2 4629  limenpsi 4652  xpnnen 7711  znnen 7714

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-pow 2818  ax-pr 2855  ax-un 3089

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-v 1858  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-pw 2459  df-sn 2470  df-pr 2471  df-op 2474  df-uni 2570  df-br 2693  df-opab 2741  df-id 2913  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-f 3275  df-f1 3276  df-fo 3277  df-f1o 3278  df-fv 3279  df-en 4509  df-dom 4510

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