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Mirrors > Home > MPE Home > Th. List > csbpredg | Structured version Visualization version GIF version |
Description: Move class substitution in and out of the predecessor class of a relation. (Contributed by ML, 25-Oct-2020.) |
Ref | Expression |
---|---|
csbpredg | ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌Pred(𝑅, 𝐷, 𝑋) = Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, ⦋𝐴 / 𝑥⦌𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbin 4448 | . . 3 ⊢ ⦋𝐴 / 𝑥⦌(𝐷 ∩ (◡𝑅 “ {𝑋})) = (⦋𝐴 / 𝑥⦌𝐷 ∩ ⦋𝐴 / 𝑥⦌(◡𝑅 “ {𝑋})) | |
2 | csbima12 6099 | . . . . 5 ⊢ ⦋𝐴 / 𝑥⦌(◡𝑅 “ {𝑋}) = (⦋𝐴 / 𝑥⦌◡𝑅 “ ⦋𝐴 / 𝑥⦌{𝑋}) | |
3 | csbcnv 5897 | . . . . . . 7 ⊢ ◡⦋𝐴 / 𝑥⦌𝑅 = ⦋𝐴 / 𝑥⦌◡𝑅 | |
4 | 3 | imaeq1i 6077 | . . . . . 6 ⊢ (◡⦋𝐴 / 𝑥⦌𝑅 “ ⦋𝐴 / 𝑥⦌{𝑋}) = (⦋𝐴 / 𝑥⦌◡𝑅 “ ⦋𝐴 / 𝑥⦌{𝑋}) |
5 | csbsng 4713 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝑋} = {⦋𝐴 / 𝑥⦌𝑋}) | |
6 | 5 | imaeq2d 6080 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (◡⦋𝐴 / 𝑥⦌𝑅 “ ⦋𝐴 / 𝑥⦌{𝑋}) = (◡⦋𝐴 / 𝑥⦌𝑅 “ {⦋𝐴 / 𝑥⦌𝑋})) |
7 | 4, 6 | eqtr3id 2789 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌◡𝑅 “ ⦋𝐴 / 𝑥⦌{𝑋}) = (◡⦋𝐴 / 𝑥⦌𝑅 “ {⦋𝐴 / 𝑥⦌𝑋})) |
8 | 2, 7 | eqtrid 2787 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(◡𝑅 “ {𝑋}) = (◡⦋𝐴 / 𝑥⦌𝑅 “ {⦋𝐴 / 𝑥⦌𝑋})) |
9 | 8 | ineq2d 4228 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝐷 ∩ ⦋𝐴 / 𝑥⦌(◡𝑅 “ {𝑋})) = (⦋𝐴 / 𝑥⦌𝐷 ∩ (◡⦋𝐴 / 𝑥⦌𝑅 “ {⦋𝐴 / 𝑥⦌𝑋}))) |
10 | 1, 9 | eqtrid 2787 | . 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐷 ∩ (◡𝑅 “ {𝑋})) = (⦋𝐴 / 𝑥⦌𝐷 ∩ (◡⦋𝐴 / 𝑥⦌𝑅 “ {⦋𝐴 / 𝑥⦌𝑋}))) |
11 | df-pred 6323 | . . 3 ⊢ Pred(𝑅, 𝐷, 𝑋) = (𝐷 ∩ (◡𝑅 “ {𝑋})) | |
12 | 11 | csbeq2i 3916 | . 2 ⊢ ⦋𝐴 / 𝑥⦌Pred(𝑅, 𝐷, 𝑋) = ⦋𝐴 / 𝑥⦌(𝐷 ∩ (◡𝑅 “ {𝑋})) |
13 | df-pred 6323 | . 2 ⊢ Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, ⦋𝐴 / 𝑥⦌𝑋) = (⦋𝐴 / 𝑥⦌𝐷 ∩ (◡⦋𝐴 / 𝑥⦌𝑅 “ {⦋𝐴 / 𝑥⦌𝑋})) | |
14 | 10, 12, 13 | 3eqtr4g 2800 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌Pred(𝑅, 𝐷, 𝑋) = Pred(⦋𝐴 / 𝑥⦌𝑅, ⦋𝐴 / 𝑥⦌𝐷, ⦋𝐴 / 𝑥⦌𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ⦋csb 3908 ∩ cin 3962 {csn 4631 ◡ccnv 5688 “ cima 5692 Predcpred 6322 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 |
This theorem is referenced by: csbfrecsg 8308 |
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