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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 3185 | . . 3 ⊢ (𝐶 ∈ (𝐹 “ 𝐵) → 𝐶 ∈ V) | |
2 | 1 | anim2i 591 | . 2 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝐶 ∈ (𝐹 “ 𝐵)) → ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝐶 ∈ V)) |
3 | fvex 6113 | . . . . 5 ⊢ (𝐹‘𝑥) ∈ V | |
4 | eleq1 2676 | . . . . 5 ⊢ ((𝐹‘𝑥) = 𝐶 → ((𝐹‘𝑥) ∈ V ↔ 𝐶 ∈ V)) | |
5 | 3, 4 | mpbii 222 | . . . 4 ⊢ ((𝐹‘𝑥) = 𝐶 → 𝐶 ∈ V) |
6 | 5 | rexlimivw 3011 | . . 3 ⊢ (∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶 → 𝐶 ∈ V) |
7 | 6 | anim2i 591 | . 2 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶) → ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝐶 ∈ V)) |
8 | eleq1 2676 | . . . . . 6 ⊢ (𝑦 = 𝐶 → (𝑦 ∈ (𝐹 “ 𝐵) ↔ 𝐶 ∈ (𝐹 “ 𝐵))) | |
9 | eqeq2 2621 | . . . . . . 7 ⊢ (𝑦 = 𝐶 → ((𝐹‘𝑥) = 𝑦 ↔ (𝐹‘𝑥) = 𝐶)) | |
10 | 9 | rexbidv 3034 | . . . . . 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 5902 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹) | |
14 | fndm 5904 | . . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
15 | 14 | sseq2d 3596 | . . . . . . 7 ⊢ (𝐹 Fn 𝐴 → (𝐵 ⊆ dom 𝐹 ↔ 𝐵 ⊆ 𝐴)) |
16 | 15 | biimpar 501 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ dom 𝐹) |
17 | dfimafn 6155 | . . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ dom 𝐹) → (𝐹 “ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦}) | |
18 | 13, 16, 17 | syl2an2r 872 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐹 “ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦}) |
19 | 18 | abeq2d 2721 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦)) |
20 | 12, 19 | vtoclg 3239 | . . 3 ⊢ (𝐶 ∈ V → ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶))) |
21 | 20 | impcom 445 | . 2 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝐶 ∈ V) → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶)) |
22 | 2, 7, 21 | pm5.21nd 939 | 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 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {cab 2596 ∃wrex 2897 Vcvv 3173 ⊆ 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: fvelimabd 6164 ssimaex 6173 rexima 6401 ralima 6402 f1elima 6421 ovelimab 6710 tcrank 8630 ackbij2 8948 fin1a2lem6 9110 iunfo 9240 grothomex 9530 axpre-sup 9869 injresinjlem 12450 lmhmima 18868 txkgen 21265 fmucndlem 21905 mdegldg 23630 ig1peu 23735 efopn 24204 cusgrares 26001 pjimai 28419 fimarab 28825 fimaproj 29228 qtophaus 29231 indf1ofs 29415 eulerpartgbij 29761 eulerpartlemgvv 29765 ballotlemsima 29904 elmthm 30727 nocvxmin 31090 isnacs2 36287 isnacs3 36291 islmodfg 36657 kercvrlsm 36671 isnumbasgrplem2 36693 dfacbasgrp 36697 unima 38340 fourierdlem62 39061 |
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