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Mirrors > Home > MPE Home > Th. List > fvelimab | Structured version Visualization version GIF version |
Description: Function value in an image. (Contributed by NM, 20-Jan-2007.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by David Abernethy, 17-Dec-2011.) |
Ref | Expression |
---|---|
fvelimab | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3364 | . . 3 ⊢ (𝐶 ∈ (𝐹 “ 𝐵) → 𝐶 ∈ V) | |
2 | 1 | anim2i 603 | . 2 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝐶 ∈ (𝐹 “ 𝐵)) → ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝐶 ∈ V)) |
3 | fvex 6344 | . . . . 5 ⊢ (𝐹‘𝑥) ∈ V | |
4 | eleq1 2838 | . . . . 5 ⊢ ((𝐹‘𝑥) = 𝐶 → ((𝐹‘𝑥) ∈ V ↔ 𝐶 ∈ V)) | |
5 | 3, 4 | mpbii 223 | . . . 4 ⊢ ((𝐹‘𝑥) = 𝐶 → 𝐶 ∈ V) |
6 | 5 | rexlimivw 3177 | . . 3 ⊢ (∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶 → 𝐶 ∈ V) |
7 | 6 | anim2i 603 | . 2 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶) → ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝐶 ∈ V)) |
8 | eleq1 2838 | . . . . . 6 ⊢ (𝑦 = 𝐶 → (𝑦 ∈ (𝐹 “ 𝐵) ↔ 𝐶 ∈ (𝐹 “ 𝐵))) | |
9 | eqeq2 2782 | . . . . . . 7 ⊢ (𝑦 = 𝐶 → ((𝐹‘𝑥) = 𝑦 ↔ (𝐹‘𝑥) = 𝐶)) | |
10 | 9 | rexbidv 3200 | . . . . . 6 ⊢ (𝑦 = 𝐶 → (∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦 ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶)) |
11 | 8, 10 | bibi12d 334 | . . . . 5 ⊢ (𝑦 = 𝐶 → ((𝑦 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦) ↔ (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶))) |
12 | 11 | imbi2d 329 | . . . 4 ⊢ (𝑦 = 𝐶 → (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦)) ↔ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶)))) |
13 | fnfun 6127 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹) | |
14 | fndm 6129 | . . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
15 | 14 | sseq2d 3782 | . . . . . . 7 ⊢ (𝐹 Fn 𝐴 → (𝐵 ⊆ dom 𝐹 ↔ 𝐵 ⊆ 𝐴)) |
16 | 15 | biimpar 463 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ dom 𝐹) |
17 | dfimafn 6389 | . . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ dom 𝐹) → (𝐹 “ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦}) | |
18 | 13, 16, 17 | syl2an2r 664 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐹 “ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦}) |
19 | 18 | abeq2d 2883 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦)) |
20 | 12, 19 | vtoclg 3417 | . . 3 ⊢ (𝐶 ∈ V → ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶))) |
21 | 20 | impcom 394 | . 2 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝐶 ∈ V) → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶)) |
22 | 2, 7, 21 | pm5.21nd 803 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 {cab 2757 ∃wrex 3062 Vcvv 3351 ⊆ wss 3723 dom cdm 5250 “ cima 5253 Fun wfun 6024 Fn wfn 6025 ‘cfv 6030 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pr 5035 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4227 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-br 4788 df-opab 4848 df-id 5158 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-iota 5993 df-fun 6032 df-fn 6033 df-fv 6038 |
This theorem is referenced by: fvelimabd 6398 ssimaex 6407 rexima 6642 ralima 6643 f1elima 6665 ovelimab 6962 tcrank 8914 djuun 8955 ackbij2 9270 fin1a2lem6 9432 iunfo 9566 grothomex 9856 axpre-sup 10195 injresinjlem 12795 lmhmima 19259 txkgen 21675 fmucndlem 22314 mdegldg 24045 ig1peu 24150 efopn 24624 pjimai 29374 fimarab 29784 fimaproj 30239 qtophaus 30242 indf1ofs 30427 eulerpartgbij 30773 eulerpartlemgvv 30777 ballotlemsima 30916 elmthm 31810 nocvxmin 32230 isnacs2 37795 isnacs3 37799 islmodfg 38165 kercvrlsm 38179 isnumbasgrplem2 38200 dfacbasgrp 38204 unima 39865 fourierdlem62 40899 |
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