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| Mirrors > Home > MPE Home > Th. List > fnfvima | Structured version Visualization version GIF version | ||
| 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.) |
| Ref | Expression |
|---|---|
| fnfvima | ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋) ∈ (𝐹 “ 𝑆)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fnfun 6668 | . . . 4 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹) | |
| 2 | 1 | 3ad2ant1 1134 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → Fun 𝐹) |
| 3 | simp2 1138 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → 𝑆 ⊆ 𝐴) | |
| 4 | fndm 6671 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
| 5 | 4 | 3ad2ant1 1134 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → dom 𝐹 = 𝐴) |
| 6 | 3, 5 | sseqtrrd 4021 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → 𝑆 ⊆ dom 𝐹) |
| 7 | 2, 6 | jca 511 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → (Fun 𝐹 ∧ 𝑆 ⊆ dom 𝐹)) |
| 8 | simp3 1139 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆) | |
| 9 | funfvima2 7251 | . 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)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 dom cdm 5685 “ cima 5688 Fun wfun 6555 Fn wfn 6556 ‘cfv 6561 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 |
| This theorem is referenced by: fnfvimad 7254 isomin 7357 isofrlem 7360 fnwelem 8156 fimaproj 8160 php3 9249 php3OLD 9261 fissuni 9397 unxpwdom2 9628 cantnflt 9712 dfac12lem2 10185 ackbij2 10282 isf34lem7 10419 isf34lem6 10420 zorn2lem2 10537 ttukeylem5 10553 tskuni 10823 axpre-sup 11209 limsupval2 15516 mgmhmima 18728 mhmimalem 18837 mhmima 18838 ghmnsgima 19258 psgnunilem1 19511 dprdfeq0 20042 dprd2dlem1 20061 rhmimasubrnglem 20565 lmhmima 21046 lmcnp 23312 basqtop 23719 tgqtop 23720 kqfvima 23738 reghmph 23801 uzrest 23905 qustgpopn 24128 qustgplem 24129 cphsqrtcl 25218 lhop 26055 ig1peu 26214 ig1pdvds 26219 plypf1 26251 nosupno 27748 nosupbday 27750 noinfno 27763 noinfbday 27765 noetasuplem4 27781 noetainflem4 27785 eqscut2 27851 scutun12 27855 scutbdaybnd 27860 scutbdaybnd2 27861 scutbdaylt 27863 madebdaylemlrcut 27937 cofcut1 27954 cofcutr 27958 lrrecfr 27976 negsproplem4 28063 negsproplem5 28064 negsproplem6 28065 f1otrg 28879 txomap 33833 sitgaddlemb 34350 f1resrcmplf1dlem 35100 cvmopnlem 35283 mrsubrn 35518 msubrn 35534 poimirlem4 37631 poimirlem6 37633 poimirlem7 37634 poimirlem16 37643 poimirlem17 37644 poimirlem19 37646 poimirlem20 37647 poimirlem23 37650 cnambfre 37675 ftc1anclem7 37706 ftc1anc 37708 aks6d1c2 42131 aks6d1c7lem1 42181 isnumbasgrplem1 43113 relpmin 44973 relpfrlem 44974 funimaeq 45253 |
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