Theorem smoel2 6534

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 5455	. . . . . 6 |- ( F Fn A -> dom F = A )
2	1	eleq2d 2302	. . . . 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 6531	. . . 4 |- ( ( Smo F /\ B e. dom F /\ C e. B ) -> ( F ` C ) e. ( F ` B ) )
6	5	3expib 1233	. . 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 2203   dom cdm 4749   Fn wfn 5347   ` cfv 5352   Smo wsmo 6516

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-tr 4209  df-iord 4487  df-iota 5312  df-fn 5355  df-fv 5360  df-smo 6517

This theorem is referenced by: (None)