Theorem smoel2 6468

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 5429	. . . . . 6 |- ( F Fn A -> dom F = A )
2	1	eleq2d 2301	. . . . 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 6465	. . . 4 |- ( ( Smo F /\ B e. dom F /\ C e. B ) -> ( F ` C ) e. ( F ` B ) )
6	5	3expib 1232	. . 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 2202   dom cdm 4725   Fn wfn 5321   ` cfv 5326   Smo wsmo 6450

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-tr 4188  df-iord 4463  df-iota 5286  df-fn 5329  df-fv 5334  df-smo 6451

This theorem is referenced by: (None)