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| Mirrors > Home > MPE Home > Th. List > csbrn | Structured version Visualization version GIF version | ||
| Description: Distribute proper substitution through the range of a class. (Contributed by Alan Sare, 10-Nov-2012.) |
| Ref | Expression |
|---|---|
| csbrn | ⊢ ⦋𝐴 / 𝑥⦌ran 𝐵 = ran ⦋𝐴 / 𝑥⦌𝐵 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | csbima12 6097 | . . 3 ⊢ ⦋𝐴 / 𝑥⦌(𝐵 “ V) = (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌V) | |
| 2 | csbconstg 3918 | . . . . 5 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌V = V) | |
| 3 | 2 | imaeq2d 6078 | . . . 4 ⊢ (𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌V) = (⦋𝐴 / 𝑥⦌𝐵 “ V)) |
| 4 | 0ima 6096 | . . . . . 6 ⊢ (∅ “ V) = ∅ | |
| 5 | 4 | eqcomi 2746 | . . . . 5 ⊢ ∅ = (∅ “ V) |
| 6 | csbprc 4409 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅) | |
| 7 | 6 | imaeq1d 6077 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌V) = (∅ “ ⦋𝐴 / 𝑥⦌V)) |
| 8 | 0ima 6096 | . . . . . 6 ⊢ (∅ “ ⦋𝐴 / 𝑥⦌V) = ∅ | |
| 9 | 7, 8 | eqtrdi 2793 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌V) = ∅) |
| 10 | 6 | imaeq1d 6077 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵 “ V) = (∅ “ V)) |
| 11 | 5, 9, 10 | 3eqtr4a 2803 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌V) = (⦋𝐴 / 𝑥⦌𝐵 “ V)) |
| 12 | 3, 11 | pm2.61i 182 | . . 3 ⊢ (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌V) = (⦋𝐴 / 𝑥⦌𝐵 “ V) |
| 13 | 1, 12 | eqtri 2765 | . 2 ⊢ ⦋𝐴 / 𝑥⦌(𝐵 “ V) = (⦋𝐴 / 𝑥⦌𝐵 “ V) |
| 14 | dfrn4 6222 | . . 3 ⊢ ran 𝐵 = (𝐵 “ V) | |
| 15 | 14 | csbeq2i 3907 | . 2 ⊢ ⦋𝐴 / 𝑥⦌ran 𝐵 = ⦋𝐴 / 𝑥⦌(𝐵 “ V) |
| 16 | dfrn4 6222 | . 2 ⊢ ran ⦋𝐴 / 𝑥⦌𝐵 = (⦋𝐴 / 𝑥⦌𝐵 “ V) | |
| 17 | 13, 15, 16 | 3eqtr4i 2775 | 1 ⊢ ⦋𝐴 / 𝑥⦌ran 𝐵 = ran ⦋𝐴 / 𝑥⦌𝐵 |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⦋csb 3899 ∅c0 4333 ran crn 5686 “ cima 5688 |
| 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-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 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-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 |
| This theorem is referenced by: sbcfg 6734 csbima12gALTVD 44917 |
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