Theorem issmo 6256

Description: Conditions for which A is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 15-Nov-2011.)

Hypotheses

Ref	Expression
issmo.1	|- A : B --> On
issmo.2	|- Ord B
issmo.3	|- ( ( x e. B /\ y e. B ) -> ( x e. y -> ( A ` x ) e. ( A ` y ) ) )
issmo.4	|- dom A = B

Assertion

Ref	Expression
issmo	|- Smo A

Distinct variable group:   x, y, A

Allowed substitution hints:   B( x, y)

Proof of Theorem issmo

Step	Hyp	Ref	Expression
1		issmo.1	. . 3 |- A : B --> On
2		issmo.4	. . . 4 |- dom A = B
3	2	feq2i 5331	. . 3 |- ( A : dom A --> On <-> A : B --> On )
4	1, 3	mpbir 145	. 2 |- A : dom A --> On
5		issmo.2	. . 3 |- Ord B
6		ordeq 4350	. . . 4 |- ( dom A = B -> ( Ord dom A <-> Ord B ) )
7	2, 6	ax-mp 5	. . 3 |- ( Ord dom A <-> Ord B )
8	5, 7	mpbir 145	. 2 |- Ord dom A
9	2	eleq2i 2233	. . . 4 |- ( x e. dom A <-> x e. B )
10	2	eleq2i 2233	. . . 4 |- ( y e. dom A <-> y e. B )
11		issmo.3	. . . 4 |- ( ( x e. B /\ y e. B ) -> ( x e. y -> ( A ` x ) e. ( A ` y ) ) )
12	9, 10, 11	syl2anb 289	. . 3 |- ( ( x e. dom A /\ y e. dom A ) -> ( x e. y -> ( A ` x ) e. ( A ` y ) ) )
13	12	rgen2a 2520	. 2 |- A. x e. dom A A. y e. dom A ( x e. y -> ( A ` x ) e. ( A ` y ) )
14		df-smo 6254	. 2 |- ( Smo A <-> ( A : dom A --> On /\ Ord dom A /\ A. x e. dom A A. y e. dom A ( x e. y -> ( A ` x ) e. ( A ` y ) ) ) )
15	4, 8, 13, 14	mpbir3an 1169	1 |- Smo A

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   A.wral 2444   Ord word 4340   Oncon0 4341   dom cdm 4604   -->wf 5184   ` cfv 5188   Smo wsmo 6253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-in 3122  df-ss 3129  df-uni 3790  df-tr 4081  df-iord 4344  df-fn 5191  df-f 5192  df-smo 6254

This theorem is referenced by:  iordsmo  6265