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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 5902 | . . . 4 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹) | |
2 | 1 | 3ad2ant1 1075 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → Fun 𝐹) |
3 | simp2 1055 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → 𝑆 ⊆ 𝐴) | |
4 | fndm 5904 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
5 | 4 | 3ad2ant1 1075 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → dom 𝐹 = 𝐴) |
6 | 3, 5 | sseqtr4d 3605 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → 𝑆 ⊆ dom 𝐹) |
7 | 2, 6 | jca 553 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → (Fun 𝐹 ∧ 𝑆 ⊆ dom 𝐹)) |
8 | simp3 1056 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆) | |
9 | funfvima2 6397 | . 2 ⊢ ((Fun 𝐹 ∧ 𝑆 ⊆ dom 𝐹) → (𝑋 ∈ 𝑆 → (𝐹‘𝑋) ∈ (𝐹 “ 𝑆))) | |
10 | 7, 8, 9 | sylc 63 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋) ∈ (𝐹 “ 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 dom cdm 5038 “ cima 5041 Fun wfun 5798 Fn wfn 5799 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-fv 5812 |
This theorem is referenced by: isomin 6487 isofrlem 6490 fnwelem 7179 php3 8031 fissuni 8154 unxpwdom2 8376 cantnflt 8452 dfac12lem2 8849 ackbij2 8948 isf34lem7 9084 isf34lem6 9085 zorn2lem2 9202 ttukeylem5 9218 tskuni 9484 axpre-sup 9869 limsupval2 14059 mhmima 17186 ghmnsgima 17507 psgnunilem1 17736 dprdfeq0 18244 dprd2dlem1 18263 lmhmima 18868 lmcnp 20918 basqtop 21324 tgqtop 21325 kqfvima 21343 reghmph 21406 uzrest 21511 qustgpopn 21733 qustgplem 21734 cphsqrtcl 22792 lhop 23583 ig1peu 23735 ig1pdvds 23740 plypf1 23772 f1otrg 25551 fimaproj 29228 txomap 29229 sitgaddlemb 29737 cvmopnlem 30514 mrsubrn 30664 msubrn 30680 nobndlem8 31098 poimirlem4 32583 poimirlem6 32585 poimirlem7 32586 poimirlem16 32595 poimirlem17 32596 poimirlem19 32598 poimirlem20 32599 poimirlem23 32602 cnambfre 32628 ftc1anclem7 32661 ftc1anc 32663 isnumbasgrplem1 36690 wfximgfd 37485 mgmhmima 41592 |
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