Theorem fnfvima 6995

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 6453	. . . 4 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
2	1	3ad2ant1 1129	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → Fun 𝐹)
3		simp2 1133	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → 𝑆 ⊆ 𝐴)
4		fndm 6455	. . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
5	4	3ad2ant1 1129	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → dom 𝐹 = 𝐴)
6	3, 5	sseqtrrd 4008	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → 𝑆 ⊆ dom 𝐹)
7	2, 6	jca 514	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → (Fun 𝐹 ∧ 𝑆 ⊆ dom 𝐹))
8		simp3 1134	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆)
9		funfvima2 6993	. 2 ⊢ ((Fun 𝐹 ∧ 𝑆 ⊆ dom 𝐹) → (𝑋 ∈ 𝑆 → (𝐹‘𝑋) ∈ (𝐹 “ 𝑆)))
10	7, 8, 9	sylc 65	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋) ∈ (𝐹 “ 𝑆))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ⊆ wss 3936  dom cdm 5555   “ cima 5558  Fun wfun 6349   Fn wfn 6350  ‘cfv 6355

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-fv 6363

This theorem is referenced by:  fnfvimad  6996  isomin  7090  isofrlem  7093  fnwelem  7825  fimaproj  7829  php3  8703  fissuni  8829  unxpwdom2  9052  cantnflt  9135  dfac12lem2  9570  ackbij2  9665  isf34lem7  9801  isf34lem6  9802  zorn2lem2  9919  ttukeylem5  9935  tskuni  10205  axpre-sup  10591  limsupval2  14837  mhmima  17989  ghmnsgima  18382  psgnunilem1  18621  dprdfeq0  19144  dprd2dlem1  19163  lmhmima  19819  lmcnp  21912  basqtop  22319  tgqtop  22320  kqfvima  22338  reghmph  22401  uzrest  22505  qustgpopn  22728  qustgplem  22729  cphsqrtcl  23788  lhop  24613  ig1peu  24765  ig1pdvds  24770  plypf1  24802  f1otrg  26657  txomap  31098  sitgaddlemb  31606  f1resrcmplf1dlem  32359  cvmopnlem  32525  mrsubrn  32760  msubrn  32776  nosupno  33203  nosupbday  33205  noetalem3  33219  scutun12  33271  scutbdaybnd  33275  scutbdaylt  33276  poimirlem4  34911  poimirlem6  34913  poimirlem7  34914  poimirlem16  34923  poimirlem17  34924  poimirlem19  34926  poimirlem20  34927  poimirlem23  34930  cnambfre  34955  ftc1anclem7  34988  ftc1anc  34990  isnumbasgrplem1  39721  funimaeq  41538  mgmhmima  44089