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Mirrors > Home > MPE Home > Th. List > funimass4 | Structured version Visualization version GIF version |
Description: Membership relation for the values of a function whose image is a subclass. (Contributed by Raph Levien, 20-Nov-2006.) |
Ref | Expression |
---|---|
funimass4 | ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss2 3557 | . . 3 ⊢ ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ ∀𝑦(𝑦 ∈ (𝐹 “ 𝐴) → 𝑦 ∈ 𝐵)) | |
2 | eqcom 2617 | . . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑦) | |
3 | ssel 3562 | . . . . . . . . . . . 12 ⊢ (𝐴 ⊆ dom 𝐹 → (𝑥 ∈ 𝐴 → 𝑥 ∈ dom 𝐹)) | |
4 | funbrfvb 6148 | . . . . . . . . . . . . 13 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦)) | |
5 | 4 | ex 449 | . . . . . . . . . . . 12 ⊢ (Fun 𝐹 → (𝑥 ∈ dom 𝐹 → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦))) |
6 | 3, 5 | syl9 75 | . . . . . . . . . . 11 ⊢ (𝐴 ⊆ dom 𝐹 → (Fun 𝐹 → (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦)))) |
7 | 6 | imp31 447 | . . . . . . . . . 10 ⊢ (((𝐴 ⊆ dom 𝐹 ∧ Fun 𝐹) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦)) |
8 | 2, 7 | syl5bb 271 | . . . . . . . . 9 ⊢ (((𝐴 ⊆ dom 𝐹 ∧ Fun 𝐹) ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹‘𝑥) ↔ 𝑥𝐹𝑦)) |
9 | 8 | rexbidva 3031 | . . . . . . . 8 ⊢ ((𝐴 ⊆ dom 𝐹 ∧ Fun 𝐹) → (∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) ↔ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦)) |
10 | vex 3176 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
11 | 10 | elima 5390 | . . . . . . . 8 ⊢ (𝑦 ∈ (𝐹 “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦) |
12 | 9, 11 | syl6rbbr 278 | . . . . . . 7 ⊢ ((𝐴 ⊆ dom 𝐹 ∧ Fun 𝐹) → (𝑦 ∈ (𝐹 “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))) |
13 | 12 | imbi1d 330 | . . . . . 6 ⊢ ((𝐴 ⊆ dom 𝐹 ∧ Fun 𝐹) → ((𝑦 ∈ (𝐹 “ 𝐴) → 𝑦 ∈ 𝐵) ↔ (∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵))) |
14 | r19.23v 3005 | . . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵) ↔ (∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵)) | |
15 | 13, 14 | syl6bbr 277 | . . . . 5 ⊢ ((𝐴 ⊆ dom 𝐹 ∧ Fun 𝐹) → ((𝑦 ∈ (𝐹 “ 𝐴) → 𝑦 ∈ 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵))) |
16 | 15 | albidv 1836 | . . . 4 ⊢ ((𝐴 ⊆ dom 𝐹 ∧ Fun 𝐹) → (∀𝑦(𝑦 ∈ (𝐹 “ 𝐴) → 𝑦 ∈ 𝐵) ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵))) |
17 | ralcom4 3197 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵) ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵)) | |
18 | fvex 6113 | . . . . . . 7 ⊢ (𝐹‘𝑥) ∈ V | |
19 | eleq1 2676 | . . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑦 ∈ 𝐵 ↔ (𝐹‘𝑥) ∈ 𝐵)) | |
20 | 18, 19 | ceqsalv 3206 | . . . . . 6 ⊢ (∀𝑦(𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵) ↔ (𝐹‘𝑥) ∈ 𝐵) |
21 | 20 | ralbii 2963 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) |
22 | 17, 21 | bitr3i 265 | . . . 4 ⊢ (∀𝑦∀𝑥 ∈ 𝐴 (𝑦 = (𝐹‘𝑥) → 𝑦 ∈ 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) |
23 | 16, 22 | syl6bb 275 | . . 3 ⊢ ((𝐴 ⊆ dom 𝐹 ∧ Fun 𝐹) → (∀𝑦(𝑦 ∈ (𝐹 “ 𝐴) → 𝑦 ∈ 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) |
24 | 1, 23 | syl5bb 271 | . 2 ⊢ ((𝐴 ⊆ dom 𝐹 ∧ Fun 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) |
25 | 24 | ancoms 468 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∀wal 1473 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 ⊆ wss 3540 class class class wbr 4583 dom cdm 5038 “ cima 5041 Fun wfun 5798 ‘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: funimass3 6241 funimass5 6242 funconstss 6243 funimassov 6709 fnwelem 7179 cnfcomlem 8479 dfac12lem2 8849 ackbij1b 8944 wunom 9421 phimullem 15322 frmdss2 17223 cntzmhm2 17595 dprd2da 18264 frlmsslsp 19954 1stckgenlem 21166 txcnp 21233 ptcnplem 21234 xkopt 21268 xkoinjcn 21300 tgqtop 21325 uzrest 21511 cnflf2 21617 lmflf 21619 txflf 21620 cnextcn 21681 ghmcnp 21728 ucnima 21895 metcnp 22156 tchcph 22844 ovolficcss 23045 opnmbllem 23175 ellimc2 23447 ellimc3 23449 deg1n0ima 23653 dvloglem 24194 logf1o2 24196 dchrghm 24781 usgrares1 25939 xrofsup 28923 eulerpartlemd 29755 erdszelem2 30428 cvmlift3lem7 30561 mclsax 30720 filnetlem4 31546 poimir 32612 opnmbllem0 32615 cnres2 32732 icccncfext 38773 |
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