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| Mirrors > Home > MPE Home > Th. List > Mathboxes > csbrecsg | Structured version Visualization version GIF version | ||
| Description: Move class substitution in and out of recs. (Contributed by ML, 25-Oct-2020.) |
| Ref | Expression |
|---|---|
| csbrecsg | ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌recs(𝐹) = recs(⦋𝐴 / 𝑥⦌𝐹)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | csbwrecsg 8346 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌wrecs( E , On, 𝐹) = wrecs(⦋𝐴 / 𝑥⦌ E , ⦋𝐴 / 𝑥⦌On, ⦋𝐴 / 𝑥⦌𝐹)) | |
| 2 | csbconstg 3918 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌ E = E ) | |
| 3 | wrecseq1 8343 | . . . 4 ⊢ (⦋𝐴 / 𝑥⦌ E = E → wrecs(⦋𝐴 / 𝑥⦌ E , ⦋𝐴 / 𝑥⦌On, ⦋𝐴 / 𝑥⦌𝐹) = wrecs( E , ⦋𝐴 / 𝑥⦌On, ⦋𝐴 / 𝑥⦌𝐹)) | |
| 4 | 2, 3 | syl 17 | . . 3 ⊢ (𝐴 ∈ 𝑉 → wrecs(⦋𝐴 / 𝑥⦌ E , ⦋𝐴 / 𝑥⦌On, ⦋𝐴 / 𝑥⦌𝐹) = wrecs( E , ⦋𝐴 / 𝑥⦌On, ⦋𝐴 / 𝑥⦌𝐹)) |
| 5 | csbconstg 3918 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌On = On) | |
| 6 | wrecseq2 8344 | . . . 4 ⊢ (⦋𝐴 / 𝑥⦌On = On → wrecs( E , ⦋𝐴 / 𝑥⦌On, ⦋𝐴 / 𝑥⦌𝐹) = wrecs( E , On, ⦋𝐴 / 𝑥⦌𝐹)) | |
| 7 | 5, 6 | syl 17 | . . 3 ⊢ (𝐴 ∈ 𝑉 → wrecs( E , ⦋𝐴 / 𝑥⦌On, ⦋𝐴 / 𝑥⦌𝐹) = wrecs( E , On, ⦋𝐴 / 𝑥⦌𝐹)) |
| 8 | 1, 4, 7 | 3eqtrd 2781 | . 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌wrecs( E , On, 𝐹) = wrecs( E , On, ⦋𝐴 / 𝑥⦌𝐹)) |
| 9 | df-recs 8411 | . . 3 ⊢ recs(𝐹) = wrecs( E , On, 𝐹) | |
| 10 | 9 | csbeq2i 3907 | . 2 ⊢ ⦋𝐴 / 𝑥⦌recs(𝐹) = ⦋𝐴 / 𝑥⦌wrecs( E , On, 𝐹) |
| 11 | df-recs 8411 | . 2 ⊢ recs(⦋𝐴 / 𝑥⦌𝐹) = wrecs( E , On, ⦋𝐴 / 𝑥⦌𝐹) | |
| 12 | 8, 10, 11 | 3eqtr4g 2802 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌recs(𝐹) = recs(⦋𝐴 / 𝑥⦌𝐹)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ⦋csb 3899 E cep 5583 Oncon0 6384 wrecscwrecs 8336 recscrecs 8410 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-iota 6514 df-fv 6569 df-ov 7434 df-frecs 8306 df-wrecs 8337 df-recs 8411 |
| This theorem is referenced by: csbrdgg 37330 |
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