Theorem smoel2 7026

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 5662	. . . . . 6 |- ( F Fn A -> dom F = A )
2	1	eleq2d 2524	. . . . 5 |- ( F Fn A -> ( B e. dom F <-> B e. A ) )
3	2	anbi1d 702	. . . 4 |- ( F Fn A -> ( ( B e. dom F /\ C e. B ) <-> ( B e. A /\ C e. B ) ) )
4	3	biimprd 223	. . 3 |- ( F Fn A -> ( ( B e. A /\ C e. B ) -> ( B e. dom F /\ C e. B ) ) )
5		smoel 7023	. . . 4 |- ( ( Smo F /\ B e. dom F /\ C e. B ) -> ( F ` C ) e. ( F ` B ) )
6	5	3expib 1197	. . 3 |- ( Smo F -> ( ( B e. dom F /\ C e. B ) -> ( F ` C ) e. ( F ` B ) ) )
7	4, 6	sylan9 655	. 2 |- ( ( F Fn A /\ Smo F ) -> ( ( B e. A /\ C e. B ) -> ( F ` C ) e. ( F ` B ) ) )
8	7	imp 427	1 |- ( ( ( F Fn A /\ Smo F ) /\ ( B e. A /\ C e. B ) ) -> ( F ` C ) e. ( F ` B ) )

Syntax hints:   -> wi 4   /\ wa 367   e. wcel 1823   dom cdm 4988   Fn wfn 5565   ` cfv 5570   Smo wsmo 7008

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-tr 4533  df-ord 4870  df-iota 5534  df-fn 5573  df-fv 5578  df-smo 7009

This theorem is referenced by:  smo11  7027  smoord  7028  smogt  7030  cofsmo  8640