Theorem fnfvimad 5948

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 3452	. . 3 |- ( A i^i C ) C_ C
2		imass2 5161	. . 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 3451	. . . 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 3414	. . 3 |- ( ph -> B e. ( A i^i C ) )
10		fnfvima 5947	. . 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 1278	. 2 |- ( ph -> ( F ` B ) e. ( F " ( A i^i C ) ) )
12	3, 11	sselid 3246	1 |- ( ph -> ( F ` B ) e. ( F " C ) )

Syntax hints:   -> wi 4   e. wcel 2209   i^i cin 3219   C_ wss 3220   "cima 4775   Fn wfn 5370   ` cfv 5375

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-fv 5383

This theorem is referenced by: (None)