Theorem issmo 6346

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 5401	. . 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 4407	. . . 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 2263	. . . 4 |- ( x e. dom A <-> x e. B )
10	2	eleq2i 2263	. . . 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 2551	. 2 |- A. x e. dom A A. y e. dom A ( x e. y -> ( A ` x ) e. ( A ` y ) )
14		df-smo 6344	. 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 1181	1 |- Smo A

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   A.wral 2475   Ord word 4397   Oncon0 4398   dom cdm 4663   -->wf 5254   ` cfv 5258   Smo wsmo 6343

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-in 3163  df-ss 3170  df-uni 3840  df-tr 4132  df-iord 4401  df-fn 5261  df-f 5262  df-smo 6344

This theorem is referenced by:  iordsmo  6355