Theorem fnfvimad 5924

Description: A function's value belongs to the image. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
fnfvimad.1	|- ( ph -> F Fn A )
fnfvimad.2	|- ( ph -> B e. A )
fnfvimad.3	|- ( ph -> B e. C )

Assertion

Ref	Expression
fnfvimad	|- ( ph -> ( F ` B ) e. ( F " C ) )

Proof of Theorem fnfvimad

Step	Hyp	Ref	Expression
1		inss2 3444	. . 3 |- ( A i^i C ) C_ C
2		imass2 5140	. . 3 |- ( ( A i^i C ) C_ C -> ( F " ( A i^i C ) ) C_ ( F " C ) )
3	1, 2	ax-mp 5	. 2 |- ( F " ( A i^i C ) ) C_ ( F " C )
4		fnfvimad.1	. . 3 |- ( ph -> F Fn A )
5		inss1 3443	. . . 4 |- ( A i^i C ) C_ A
6	5	a1i 9	. . 3 |- ( ph -> ( A i^i C ) C_ A )
7		fnfvimad.2	. . . 4 |- ( ph -> B e. A )
8		fnfvimad.3	. . . 4 |- ( ph -> B e. C )
9	7, 8	elind 3406	. . 3 |- ( ph -> B e. ( A i^i C ) )
10		fnfvima 5923	. . 3 |- ( ( F Fn A /\ ( A i^i C ) C_ A /\ B e. ( A i^i C ) ) -> ( F ` B ) e. ( F " ( A i^i C ) ) )
11	4, 6, 9, 10	syl3anc 1274	. 2 |- ( ph -> ( F ` B ) e. ( F " ( A i^i C ) ) )
12	3, 11	sselid 3238	1 |- ( ph -> ( F ` B ) e. ( F " C ) )

Syntax hints:   -> wi 4   e. wcel 2205   i^i cin 3212   C_ wss 3213   "cima 4754   Fn wfn 5349   ` cfv 5354

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-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3045  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-fv 5362

This theorem is referenced by: (None)