Theorem issmo 6053

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 5155	. . 3 |- ( A : dom A --> On <-> A : B --> On )
4	1, 3	mpbir 144	. 2 |- A : dom A --> On
5		issmo.2	. . 3 |- Ord B
6		ordeq 4199	. . . 4 |- ( dom A = B -> ( Ord dom A <-> Ord B ) )
7	2, 6	ax-mp 7	. . 3 |- ( Ord dom A <-> Ord B )
8	5, 7	mpbir 144	. 2 |- Ord dom A
9	2	eleq2i 2154	. . . 4 |- ( x e. dom A <-> x e. B )
10	2	eleq2i 2154	. . . 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 285	. . 3 |- ( ( x e. dom A /\ y e. dom A ) -> ( x e. y -> ( A ` x ) e. ( A ` y ) ) )
13	12	rgen2a 2429	. 2 |- A. x e. dom A A. y e. dom A ( x e. y -> ( A ` x ) e. ( A ` y ) )
14		df-smo 6051	. 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 1125	1 |- Smo A

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1289   e. wcel 1438   A.wral 2359   Ord word 4189   Oncon0 4190   dom cdm 4438   -->wf 5011   ` cfv 5015   Smo wsmo 6050

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-in 3005  df-ss 3012  df-uni 3654  df-tr 3937  df-iord 4193  df-fn 5018  df-f 5019  df-smo 6051

This theorem is referenced by:  iordsmo  6062