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Mirrors > Home > MPE Home > Th. List > Mathboxes > pwselbasr | Structured version Visualization version GIF version |
Description: The reverse direction of pwselbasb 17430: a function between the index and base set of a structure is an element of the structure power. (Contributed by SN, 29-Jul-2024.) |
Ref | Expression |
---|---|
pwselbasr.y | ⊢ 𝑌 = (𝑅 ↑s 𝐼) |
pwselbasr.b | ⊢ 𝐵 = (Base‘𝑅) |
pwselbasr.v | ⊢ 𝑉 = (Base‘𝑌) |
pwselbasr.r | ⊢ (𝜑 → 𝑅 ∈ 𝑊) |
pwselbasr.i | ⊢ (𝜑 → 𝐼 ∈ 𝑍) |
pwselbasr.x | ⊢ (𝜑 → 𝑋:𝐼⟶𝐵) |
Ref | Expression |
---|---|
pwselbasr | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwselbasr.x | . 2 ⊢ (𝜑 → 𝑋:𝐼⟶𝐵) | |
2 | pwselbasr.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑊) | |
3 | pwselbasr.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑍) | |
4 | pwselbasr.y | . . . 4 ⊢ 𝑌 = (𝑅 ↑s 𝐼) | |
5 | pwselbasr.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
6 | pwselbasr.v | . . . 4 ⊢ 𝑉 = (Base‘𝑌) | |
7 | 4, 5, 6 | pwselbasb 17430 | . . 3 ⊢ ((𝑅 ∈ 𝑊 ∧ 𝐼 ∈ 𝑍) → (𝑋 ∈ 𝑉 ↔ 𝑋:𝐼⟶𝐵)) |
8 | 2, 3, 7 | syl2anc 583 | . 2 ⊢ (𝜑 → (𝑋 ∈ 𝑉 ↔ 𝑋:𝐼⟶𝐵)) |
9 | 1, 8 | mpbird 257 | 1 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 ⟶wf 6529 ‘cfv 6533 (class class class)co 7401 Basecbs 17140 ↑s cpws 17388 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8698 df-map 8817 df-ixp 8887 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-sup 9432 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-struct 17076 df-slot 17111 df-ndx 17123 df-base 17141 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-hom 17217 df-cco 17218 df-prds 17389 df-pws 17391 |
This theorem is referenced by: pwsgprod 41569 evlsvvval 41590 |
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