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Mirrors > Home > MPE Home > Th. List > Mathboxes > csbpredg | Structured version Visualization version GIF version |
Description: Move class substitution in and out of the predecessor class of a relationship. (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 4010 | . . 3 ⊢ ⦋𝐴 / 𝑥⦌(𝐷 ∩ (◡𝑅 “ {𝑋})) = (⦋𝐴 / 𝑥⦌𝐷 ∩ ⦋𝐴 / 𝑥⦌(◡𝑅 “ {𝑋})) | |
2 | csbima12 5483 | . . . . 5 ⊢ ⦋𝐴 / 𝑥⦌(◡𝑅 “ {𝑋}) = (⦋𝐴 / 𝑥⦌◡𝑅 “ ⦋𝐴 / 𝑥⦌{𝑋}) | |
3 | csbcnv 5306 | . . . . . . 7 ⊢ ◡⦋𝐴 / 𝑥⦌𝑅 = ⦋𝐴 / 𝑥⦌◡𝑅 | |
4 | 3 | imaeq1i 5463 | . . . . . 6 ⊢ (◡⦋𝐴 / 𝑥⦌𝑅 “ ⦋𝐴 / 𝑥⦌{𝑋}) = (⦋𝐴 / 𝑥⦌◡𝑅 “ ⦋𝐴 / 𝑥⦌{𝑋}) |
5 | csbsng 4243 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝑋} = {⦋𝐴 / 𝑥⦌𝑋}) | |
6 | 5 | imaeq2d 5466 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (◡⦋𝐴 / 𝑥⦌𝑅 “ ⦋𝐴 / 𝑥⦌{𝑋}) = (◡⦋𝐴 / 𝑥⦌𝑅 “ {⦋𝐴 / 𝑥⦌𝑋})) |
7 | 4, 6 | syl5eqr 2670 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌◡𝑅 “ ⦋𝐴 / 𝑥⦌{𝑋}) = (◡⦋𝐴 / 𝑥⦌𝑅 “ {⦋𝐴 / 𝑥⦌𝑋})) |
8 | 2, 7 | syl5eq 2668 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(◡𝑅 “ {𝑋}) = (◡⦋𝐴 / 𝑥⦌𝑅 “ {⦋𝐴 / 𝑥⦌𝑋})) |
9 | 8 | ineq2d 3814 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝐷 ∩ ⦋𝐴 / 𝑥⦌(◡𝑅 “ {𝑋})) = (⦋𝐴 / 𝑥⦌𝐷 ∩ (◡⦋𝐴 / 𝑥⦌𝑅 “ {⦋𝐴 / 𝑥⦌𝑋}))) |
10 | 1, 9 | syl5eq 2668 | . 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐷 ∩ (◡𝑅 “ {𝑋})) = (⦋𝐴 / 𝑥⦌𝐷 ∩ (◡⦋𝐴 / 𝑥⦌𝑅 “ {⦋𝐴 / 𝑥⦌𝑋}))) |
11 | df-pred 5680 | . . 3 ⊢ Pred(𝑅, 𝐷, 𝑋) = (𝐷 ∩ (◡𝑅 “ {𝑋})) | |
12 | 11 | csbeq2i 3993 | . 2 ⊢ ⦋𝐴 / 𝑥⦌Pred(𝑅, 𝐷, 𝑋) = ⦋𝐴 / 𝑥⦌(𝐷 ∩ (◡𝑅 “ {𝑋})) |
13 | df-pred 5680 | . 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 2681 | 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 1483 ∈ wcel 1990 ⦋csb 3533 ∩ cin 3573 {csn 4177 ◡ccnv 5113 “ cima 5117 Predcpred 5679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 |
This theorem is referenced by: csbwrecsg 33173 |
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