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Mirrors > Home > MPE Home > Th. List > Mathboxes > assintopval | Structured version Visualization version GIF version |
Description: The associative (closed internal binary) operations for a set. (Contributed by AV, 20-Jan-2020.) |
Ref | Expression |
---|---|
assintopval | ⊢ (𝑀 ∈ 𝑉 → ( assIntOp ‘𝑀) = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-assintop 41627 | . . 3 ⊢ assIntOp = (𝑚 ∈ V ↦ {𝑜 ∈ ( clIntOp ‘𝑚) ∣ 𝑜 assLaw 𝑚}) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝑀 ∈ 𝑉 → assIntOp = (𝑚 ∈ V ↦ {𝑜 ∈ ( clIntOp ‘𝑚) ∣ 𝑜 assLaw 𝑚})) |
3 | fveq2 6103 | . . . 4 ⊢ (𝑚 = 𝑀 → ( clIntOp ‘𝑚) = ( clIntOp ‘𝑀)) | |
4 | breq2 4587 | . . . 4 ⊢ (𝑚 = 𝑀 → (𝑜 assLaw 𝑚 ↔ 𝑜 assLaw 𝑀)) | |
5 | 3, 4 | rabeqbidv 3168 | . . 3 ⊢ (𝑚 = 𝑀 → {𝑜 ∈ ( clIntOp ‘𝑚) ∣ 𝑜 assLaw 𝑚} = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀}) |
6 | 5 | adantl 481 | . 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑚 = 𝑀) → {𝑜 ∈ ( clIntOp ‘𝑚) ∣ 𝑜 assLaw 𝑚} = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀}) |
7 | elex 3185 | . 2 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ V) | |
8 | fvex 6113 | . . . 4 ⊢ ( clIntOp ‘𝑀) ∈ V | |
9 | 8 | rabex 4740 | . . 3 ⊢ {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀} ∈ V |
10 | 9 | a1i 11 | . 2 ⊢ (𝑀 ∈ 𝑉 → {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀} ∈ V) |
11 | 2, 6, 7, 10 | fvmptd 6197 | 1 ⊢ (𝑀 ∈ 𝑉 → ( assIntOp ‘𝑀) = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 {crab 2900 Vcvv 3173 class class class wbr 4583 ↦ cmpt 4643 ‘cfv 5804 assLaw casslaw 41610 clIntOp cclintop 41623 assIntOp cassintop 41624 |
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-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
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-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-assintop 41627 |
This theorem is referenced by: assintopmap 41632 isassintop 41636 assintopcllaw 41638 |
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