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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 5111 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
3 | 1, 2 | eqtri 2761 | . . . . 5 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} |
4 | 3 | cnveqi 5717 | . . . 4 ⊢ ◡𝐹 = ◡{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} |
5 | cnvopab 5971 | . . . 4 ⊢ ◡{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {〈𝑦, 𝑥〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
6 | 4, 5 | eqtri 2761 | . . 3 ⊢ ◡𝐹 = {〈𝑦, 𝑥〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} |
7 | 6 | imaeq1i 5900 | . 2 ⊢ (◡𝐹 “ 𝐶) = ({〈𝑦, 𝑥〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} “ 𝐶) |
8 | df-ima 5538 | . . 3 ⊢ ({〈𝑦, 𝑥〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} “ 𝐶) = ran ({〈𝑦, 𝑥〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ↾ 𝐶) | |
9 | resopab 5876 | . . . . 5 ⊢ ({〈𝑦, 𝑥〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ↾ 𝐶) = {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))} | |
10 | 9 | rneqi 5780 | . . . 4 ⊢ ran ({〈𝑦, 𝑥〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ↾ 𝐶) = ran {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))} |
11 | ancom 464 | . . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝑦 ∈ 𝐶)) | |
12 | anass 472 | . . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝑦 ∈ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝑦 ∈ 𝐶))) | |
13 | 11, 12 | bitri 278 | . . . . . . . 8 ⊢ ((𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝑦 ∈ 𝐶))) |
14 | 13 | exbii 1854 | . . . . . . 7 ⊢ (∃𝑦(𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)) ↔ ∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝑦 ∈ 𝐶))) |
15 | 19.42v 1961 | . . . . . . . 8 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝑦 ∈ 𝐶))) | |
16 | dfclel 2812 | . . . . . . . . . 10 ⊢ (𝐵 ∈ 𝐶 ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝑦 ∈ 𝐶)) | |
17 | 16 | bicomi 227 | . . . . . . . . 9 ⊢ (∃𝑦(𝑦 = 𝐵 ∧ 𝑦 ∈ 𝐶) ↔ 𝐵 ∈ 𝐶) |
18 | 17 | anbi2i 626 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶)) |
19 | 15, 18 | bitri 278 | . . . . . . 7 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶)) |
20 | 14, 19 | bitri 278 | . . . . . 6 ⊢ (∃𝑦(𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶)) |
21 | 20 | abbii 2803 | . . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶)} |
22 | rnopab 5797 | . . . . 5 ⊢ ran {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))} = {𝑥 ∣ ∃𝑦(𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))} | |
23 | df-rab 3062 | . . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶)} | |
24 | 21, 22, 23 | 3eqtr4i 2771 | . . . 4 ⊢ ran {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))} = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶} |
25 | 10, 24 | eqtri 2761 | . . 3 ⊢ ran ({〈𝑦, 𝑥〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ↾ 𝐶) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶} |
26 | 8, 25 | eqtri 2761 | . 2 ⊢ ({〈𝑦, 𝑥〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} “ 𝐶) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶} |
27 | 7, 26 | eqtri 2761 | 1 ⊢ (◡𝐹 “ 𝐶) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1542 ∃wex 1786 ∈ wcel 2114 {cab 2716 {crab 3057 {copab 5092 ↦ cmpt 5110 ◡ccnv 5524 ran crn 5526 ↾ cres 5527 “ cima 5528 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-rab 3062 df-v 3400 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-br 5031 df-opab 5093 df-mpt 5111 df-xp 5531 df-rel 5532 df-cnv 5533 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 |
This theorem is referenced by: mptiniseg 6071 dmmpt 6072 fmpt 6884 f1oresrab 6899 mptsuppdifd 7881 r0weon 9512 compss 9876 infrenegsup 11701 eqglact 18449 odngen 18820 pjdm 20523 psrbagsn 20875 coe1mul2lem2 21043 xkoccn 22370 txcnmpt 22375 txdis1cn 22386 pthaus 22389 txkgen 22403 xkoco1cn 22408 xkoco2cn 22409 xkoinjcn 22438 txconn 22440 imasnopn 22441 imasncld 22442 imasncls 22443 ptcmplem1 22803 ptcmplem3 22805 ptcmplem4 22806 tmdgsum2 22847 symgtgp 22857 tgpconncompeqg 22863 ghmcnp 22866 tgpt0 22870 qustgpopn 22871 qustgphaus 22874 eltsms 22884 prdsxmslem2 23282 efopn 25401 atansopn 25670 xrlimcnp 25706 fpwrelmapffslem 30642 ptrest 35399 mbfposadd 35447 cnambfre 35448 itg2addnclem2 35452 iblabsnclem 35463 ftc1anclem1 35473 ftc1anclem6 35478 pwfi2f1o 40493 smfpimioo 43860 |
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