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Mirrors > Home > MPE Home > Th. List > mptpreima | Structured version Visualization version GIF version |
Description: The preimage of a function in maps-to notation. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
Ref | Expression |
---|---|
dmmpt.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
Ref | Expression |
---|---|
mptpreima | ⊢ (◡𝐹 “ 𝐶) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmmpt.1 | . . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | df-mpt 5158 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
3 | 1, 2 | eqtri 2766 | . . . . 5 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} |
4 | 3 | cnveqi 5783 | . . . 4 ⊢ ◡𝐹 = ◡{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} |
5 | cnvopab 6042 | . . . 4 ⊢ ◡{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {〈𝑦, 𝑥〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
6 | 4, 5 | eqtri 2766 | . . 3 ⊢ ◡𝐹 = {〈𝑦, 𝑥〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} |
7 | 6 | imaeq1i 5966 | . 2 ⊢ (◡𝐹 “ 𝐶) = ({〈𝑦, 𝑥〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} “ 𝐶) |
8 | df-ima 5602 | . . 3 ⊢ ({〈𝑦, 𝑥〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} “ 𝐶) = ran ({〈𝑦, 𝑥〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ↾ 𝐶) | |
9 | resopab 5942 | . . . . 5 ⊢ ({〈𝑦, 𝑥〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ↾ 𝐶) = {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))} | |
10 | 9 | rneqi 5846 | . . . 4 ⊢ ran ({〈𝑦, 𝑥〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ↾ 𝐶) = ran {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))} |
11 | ancom 461 | . . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝑦 ∈ 𝐶)) | |
12 | anass 469 | . . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝑦 ∈ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝑦 ∈ 𝐶))) | |
13 | 11, 12 | bitri 274 | . . . . . . . 8 ⊢ ((𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝑦 ∈ 𝐶))) |
14 | 13 | exbii 1850 | . . . . . . 7 ⊢ (∃𝑦(𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)) ↔ ∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝑦 ∈ 𝐶))) |
15 | 19.42v 1957 | . . . . . . . 8 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝑦 ∈ 𝐶))) | |
16 | dfclel 2817 | . . . . . . . . . 10 ⊢ (𝐵 ∈ 𝐶 ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝑦 ∈ 𝐶)) | |
17 | 16 | bicomi 223 | . . . . . . . . 9 ⊢ (∃𝑦(𝑦 = 𝐵 ∧ 𝑦 ∈ 𝐶) ↔ 𝐵 ∈ 𝐶) |
18 | 17 | anbi2i 623 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶)) |
19 | 15, 18 | bitri 274 | . . . . . . 7 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶)) |
20 | 14, 19 | bitri 274 | . . . . . 6 ⊢ (∃𝑦(𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶)) |
21 | 20 | abbii 2808 | . . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶)} |
22 | rnopab 5863 | . . . . 5 ⊢ ran {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))} = {𝑥 ∣ ∃𝑦(𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))} | |
23 | df-rab 3073 | . . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶)} | |
24 | 21, 22, 23 | 3eqtr4i 2776 | . . . 4 ⊢ ran {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))} = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶} |
25 | 10, 24 | eqtri 2766 | . . 3 ⊢ ran ({〈𝑦, 𝑥〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ↾ 𝐶) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶} |
26 | 8, 25 | eqtri 2766 | . 2 ⊢ ({〈𝑦, 𝑥〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} “ 𝐶) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶} |
27 | 7, 26 | eqtri 2766 | 1 ⊢ (◡𝐹 “ 𝐶) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1539 ∃wex 1782 ∈ wcel 2106 {cab 2715 {crab 3068 {copab 5136 ↦ cmpt 5157 ◡ccnv 5588 ran crn 5590 ↾ cres 5591 “ cima 5592 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-br 5075 df-opab 5137 df-mpt 5158 df-xp 5595 df-rel 5596 df-cnv 5597 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 |
This theorem is referenced by: mptiniseg 6142 dmmpt 6143 fmpt 6984 f1oresrab 6999 mptsuppdifd 8002 r0weon 9768 compss 10132 infrenegsup 11958 eqglact 18807 odngen 19182 pjdm 20914 psrbagsn 21271 coe1mul2lem2 21439 xkoccn 22770 txcnmpt 22775 txdis1cn 22786 pthaus 22789 txkgen 22803 xkoco1cn 22808 xkoco2cn 22809 xkoinjcn 22838 txconn 22840 imasnopn 22841 imasncld 22842 imasncls 22843 ptcmplem1 23203 ptcmplem3 23205 ptcmplem4 23206 tmdgsum2 23247 symgtgp 23257 tgpconncompeqg 23263 ghmcnp 23266 tgpt0 23270 qustgpopn 23271 qustgphaus 23274 eltsms 23284 prdsxmslem2 23685 efopn 25813 atansopn 26082 xrlimcnp 26118 fpwrelmapffslem 31067 ptrest 35776 mbfposadd 35824 cnambfre 35825 itg2addnclem2 35829 iblabsnclem 35840 ftc1anclem1 35850 ftc1anclem6 35855 pwfi2f1o 40921 smfpimioo 44321 |
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