Theorem fnfvima 5894

Description: The function value of an operand in a set is contained in the image of that set, using the Fn abbreviation. (Contributed by Stefan O'Rear, 10-Mar-2015.)

Assertion

Ref	Expression
fnfvima	⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋) ∈ (𝐹 “ 𝑆))

Proof of Theorem fnfvima

Step	Hyp	Ref	Expression
1		fnfun 5429	. . . 4 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
2	1	3ad2ant1 1044	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → Fun 𝐹)
3		simp2 1024	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → 𝑆 ⊆ 𝐴)
4		fndm 5431	. . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
5	4	3ad2ant1 1044	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → dom 𝐹 = 𝐴)
6	3, 5	sseqtrrd 3265	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → 𝑆 ⊆ dom 𝐹)
7	2, 6	jca 306	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → (Fun 𝐹 ∧ 𝑆 ⊆ dom 𝐹))
8		simp3 1025	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆)
9		funfvima2 5892	. 2 ⊢ ((Fun 𝐹 ∧ 𝑆 ⊆ dom 𝐹) → (𝑋 ∈ 𝑆 → (𝐹‘𝑋) ∈ (𝐹 “ 𝑆)))
10	7, 8, 9	sylc 62	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋) ∈ (𝐹 “ 𝑆))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1004   = wceq 1397   ∈ wcel 2201   ⊆ wss 3199  dom cdm 4727   “ cima 4730  Fun wfun 5322   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:  fnfvimad  5895  iseqf1olemnab  10769  ennnfonelemrn  13063  mhmima  13597  ghmnsgima  13878  lmtopcnp  15003