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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 4249 | . 2 ⊢ (𝐴 ∈ (𝐵 ↑m 𝐶) → ¬ (𝐵 ↑m 𝐶) = ∅) | |
2 | fnmap 8396 | . . . 4 ⊢ ↑m Fn (V × V) | |
3 | 2 | fndmi 6426 | . . 3 ⊢ dom ↑m = (V × V) |
4 | 3 | ndmov 7312 | . 2 ⊢ (¬ (𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐵 ↑m 𝐶) = ∅) |
5 | 1, 4 | nsyl2 143 | 1 ⊢ (𝐴 ∈ (𝐵 ↑m 𝐶) → (𝐵 ∈ V ∧ 𝐶 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 × cxp 5517 (class class class)co 7135 ↑m cmap 8389 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-map 8391 |
This theorem is referenced by: elmapi 8411 elmapssres 8414 mapsspm 8423 elmapresaun 8427 mapss 8436 ralxpmap 8443 mapdom1 8666 wemapwe 9144 isf34lem6 9791 mndvcl 20998 mndvass 20999 mndvlid 21000 mndvrid 21001 grpvlinv 21002 grpvrinv 21003 mhmvlin 21004 tposmap 21062 satfv1lem 32722 mapfzcons 39657 ovnhoilem2 43241 |
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