Theorem smoel2 6356

Description: A strictly monotone ordinal function preserves the epsilon relation. (Contributed by Mario Carneiro, 12-Mar-2013.)

Assertion

Ref	Expression
smoel2	|- ( ( ( F Fn A /\ Smo F ) /\ ( B e. A /\ C e. B ) ) -> ( F ` C ) e. ( F ` B ) )

Proof of Theorem smoel2

Step	Hyp	Ref	Expression
1		fndm 5353	. . . . . 6 |- ( F Fn A -> dom F = A )
2	1	eleq2d 2263	. . . . 5 |- ( F Fn A -> ( B e. dom F <-> B e. A ) )
3	2	anbi1d 465	. . . 4 |- ( F Fn A -> ( ( B e. dom F /\ C e. B ) <-> ( B e. A /\ C e. B ) ) )
4	3	biimprd 158	. . 3 |- ( F Fn A -> ( ( B e. A /\ C e. B ) -> ( B e. dom F /\ C e. B ) ) )
5		smoel 6353	. . . 4 |- ( ( Smo F /\ B e. dom F /\ C e. B ) -> ( F ` C ) e. ( F ` B ) )
6	5	3expib 1208	. . 3 |- ( Smo F -> ( ( B e. dom F /\ C e. B ) -> ( F ` C ) e. ( F ` B ) ) )
7	4, 6	sylan9 409	. 2 |- ( ( F Fn A /\ Smo F ) -> ( ( B e. A /\ C e. B ) -> ( F ` C ) e. ( F ` B ) ) )
8	7	imp 124	1 |- ( ( ( F Fn A /\ Smo F ) /\ ( B e. A /\ C e. B ) ) -> ( F ` C ) e. ( F ` B ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2164   dom cdm 4659   Fn wfn 5249   ` cfv 5254   Smo wsmo 6338

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-tr 4128  df-iord 4397  df-iota 5215  df-fn 5257  df-fv 5262  df-smo 6339

This theorem is referenced by: (None)