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Mirrors > Home > MPE Home > Th. List > Mathboxes > pmapssbaN | Structured version Visualization version GIF version |
Description: A weakening of pmapssat 36889 to shorten some proofs. (Contributed by NM, 7-Mar-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pmapssba.b | ⊢ 𝐵 = (Base‘𝐾) |
pmapssba.m | ⊢ 𝑀 = (pmap‘𝐾) |
Ref | Expression |
---|---|
pmapssbaN | ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pmapssba.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | eqid 2821 | . . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
3 | pmapssba.m | . . 3 ⊢ 𝑀 = (pmap‘𝐾) | |
4 | 1, 2, 3 | pmapssat 36889 | . 2 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) ⊆ (Atoms‘𝐾)) |
5 | 1, 2 | atssbase 36420 | . 2 ⊢ (Atoms‘𝐾) ⊆ 𝐵 |
6 | 4, 5 | sstrdi 3978 | 1 ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 ‘cfv 6349 Basecbs 16477 Atomscatm 36393 pmapcpmap 36627 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ats 36397 df-pmap 36634 |
This theorem is referenced by: paddunN 37057 |
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