Theorem fnfvimad 5895

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 3427	. . 3 |- ( A i^i C ) C_ C
2		imass2 5114	. . 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 3426	. . . 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 3391	. . 3 |- ( ph -> B e. ( A i^i C ) )
10		fnfvima 5894	. . 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 1273	. 2 |- ( ph -> ( F ` B ) e. ( F " ( A i^i C ) ) )
12	3, 11	sselid 3224	1 |- ( ph -> ( F ` B ) e. ( F " C ) )

Syntax hints:   -> wi 4   e. wcel 2201   i^i cin 3198   C_ wss 3199   "cima 4730   Fn wfn 5323   ` cfv 5328

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-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-v 2803  df-sbc 3031  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-br 4090  df-opab 4152  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-fv 5336

This theorem is referenced by: (None)