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Mirrors > Home > MPE Home > Th. List > csbco | Structured version Visualization version GIF version |
Description: Composition law for chained substitutions into a class. (Contributed by NM, 10-Nov-2005.) |
Ref | Expression |
---|---|
csbco | ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-csb 3500 | . . . . . 6 ⊢ ⦋𝑦 / 𝑥⦌𝐵 = {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝐵} | |
2 | 1 | abeq2i 2722 | . . . . 5 ⊢ (𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵 ↔ [𝑦 / 𝑥]𝑧 ∈ 𝐵) |
3 | 2 | sbcbii 3458 | . . . 4 ⊢ ([𝐴 / 𝑦]𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵 ↔ [𝐴 / 𝑦][𝑦 / 𝑥]𝑧 ∈ 𝐵) |
4 | sbcco 3425 | . . . 4 ⊢ ([𝐴 / 𝑦][𝑦 / 𝑥]𝑧 ∈ 𝐵 ↔ [𝐴 / 𝑥]𝑧 ∈ 𝐵) | |
5 | 3, 4 | bitri 263 | . . 3 ⊢ ([𝐴 / 𝑦]𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵 ↔ [𝐴 / 𝑥]𝑧 ∈ 𝐵) |
6 | 5 | abbii 2726 | . 2 ⊢ {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵} = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐵} |
7 | df-csb 3500 | . 2 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐵 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵} | |
8 | df-csb 3500 | . 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐵} | |
9 | 6, 7, 8 | 3eqtr4i 2642 | 1 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 {cab 2596 [wsbc 3402 ⦋csb 3499 |
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-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-v 3175 df-sbc 3403 df-csb 3500 |
This theorem is referenced by: csbnest1g 3953 csbvarg 3955 fvmpt2curryd 7284 zsum 14296 fsum 14298 zprod 14506 fprod 14510 gsumply1eq 19496 bj-csbsn 32091 sbccom2 33100 disjrnmpt2 38370 disjinfi 38375 fsumsplitf 38634 dvmptmulf 38827 dvmptfprod 38835 |
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