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Mirrors > Home > MPE Home > Th. List > elmapex | Structured version Visualization version GIF version |
Description: Eliminate antecedent for mapping theorems: domain can be taken to be a set. (Contributed by Stefan O'Rear, 8-Oct-2014.) |
Ref | Expression |
---|---|
elmapex | ⊢ (𝐴 ∈ (𝐵 ↑m 𝐶) → (𝐵 ∈ V ∧ 𝐶 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0i 4346 | . 2 ⊢ (𝐴 ∈ (𝐵 ↑m 𝐶) → ¬ (𝐵 ↑m 𝐶) = ∅) | |
2 | fnmap 8872 | . . . 4 ⊢ ↑m Fn (V × V) | |
3 | 2 | fndmi 6673 | . . 3 ⊢ dom ↑m = (V × V) |
4 | 3 | ndmov 7617 | . 2 ⊢ (¬ (𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐵 ↑m 𝐶) = ∅) |
5 | 1, 4 | nsyl2 141 | 1 ⊢ (𝐴 ∈ (𝐵 ↑m 𝐶) → (𝐵 ∈ V ∧ 𝐶 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 × cxp 5687 (class class class)co 7431 ↑m cmap 8865 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-map 8867 |
This theorem is referenced by: elmapi 8888 elmapssres 8906 mapsspm 8915 elmapresaun 8919 mapss 8928 ralxpmap 8935 mapdom1 9181 wemapwe 9735 isf34lem6 10418 mndvcl 18823 mndvass 18824 mndvlid 18825 mndvrid 18826 mhmvlin 18827 grpvlinv 22418 grpvrinv 22419 tposmap 22479 satfv1lem 35347 mapcod 42263 mapfzcons 42704 ovnhoilem2 46558 |
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