Theorem issmo 6397

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 5439	. . 3 |- ( A : dom A --> On <-> A : B --> On )
4	1, 3	mpbir 146	. 2 |- A : dom A --> On
5		issmo.2	. . 3 |- Ord B
6		ordeq 4437	. . . 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 146	. 2 |- Ord dom A
9	2	eleq2i 2274	. . . 4 |- ( x e. dom A <-> x e. B )
10	2	eleq2i 2274	. . . 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 291	. . 3 |- ( ( x e. dom A /\ y e. dom A ) -> ( x e. y -> ( A ` x ) e. ( A ` y ) ) )
13	12	rgen2a 2562	. 2 |- A. x e. dom A A. y e. dom A ( x e. y -> ( A ` x ) e. ( A ` y ) )
14		df-smo 6395	. 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 1182	1 |- Smo A

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2178   A.wral 2486   Ord word 4427   Oncon0 4428   dom cdm 4693   -->wf 5286   ` cfv 5290   Smo wsmo 6394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-in 3180  df-ss 3187  df-uni 3865  df-tr 4159  df-iord 4431  df-fn 5293  df-f 5294  df-smo 6395

This theorem is referenced by:  iordsmo  6406