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Mirrors > Home > MPE Home > Th. List > elmapd | Structured version Visualization version GIF version |
Description: Deduction form of elmapg 7757. (Contributed by BJ, 11-Apr-2020.) |
Ref | Expression |
---|---|
elmapd.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
elmapd.b | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
Ref | Expression |
---|---|
elmapd | ⊢ (𝜑 → (𝐶 ∈ (𝐴 ↑𝑚 𝐵) ↔ 𝐶:𝐵⟶𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elmapd.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | elmapd.b | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
3 | elmapg 7757 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐶 ∈ (𝐴 ↑𝑚 𝐵) ↔ 𝐶:𝐵⟶𝐴)) | |
4 | 1, 2, 3 | syl2anc 691 | 1 ⊢ (𝜑 → (𝐶 ∈ (𝐴 ↑𝑚 𝐵) ↔ 𝐶:𝐵⟶𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∈ wcel 1977 ⟶wf 5800 (class class class)co 6549 ↑𝑚 cmap 7744 |
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-8 1979 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-pow 4769 ax-pr 4833 ax-un 6847 |
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-pw 4110 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-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 |
This theorem is referenced by: elmapssres 7768 mapss 7786 ralxpmap 7793 mapen 8009 mapunen 8014 f1finf1o 8072 mapfienlem3 8195 mapfien 8196 cantnfs 8446 acni 8751 infmap2 8923 fin23lem32 9049 iundom2g 9241 wunf 9428 hashf1lem1 13096 hashf1lem2 13097 prdsplusg 15941 prdsmulr 15942 prdsvsca 15943 elsetchom 16554 setcco 16556 isga 17547 evls1sca 19509 mamures 20015 mat1dimmul 20101 1mavmul 20173 mdetunilem9 20245 cnpdis 20907 xkopjcn 21269 indishmph 21411 tsmsxplem2 21767 dchrfi 24780 fcobij 28888 mbfmcst 29648 1stmbfm 29649 2ndmbfm 29650 mbfmco 29653 sibfof 29729 mapco2g 36295 elmapresaun 36352 rfovcnvf1od 37318 fsovfd 37326 fsovcnvlem 37327 dssmapnvod 37334 clsk3nimkb 37358 ntrelmap 37443 clselmap 37445 k0004lem2 37466 elmapsnd 38391 mapss2 38392 unirnmap 38395 inmap 38396 difmapsn 38399 unirnmapsn 38401 dvnprodlem1 38836 fourierdlem14 39014 fourierdlem15 39015 fourierdlem81 39080 fourierdlem92 39091 rrnprjdstle 39197 subsaliuncllem 39251 hoidmvlelem3 39487 ovolval2lem 39533 ovolval4lem2 39540 ovolval5lem2 39543 ovnovollem1 39546 smfmullem4 39679 el0ldep 42049 |
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