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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 6082 | . . 3 ⊢ ⦋𝐴 / 𝑥⦌(𝐵 “ V) = (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌V) | |
2 | csbconstg 3911 | . . . . 5 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌V = V) | |
3 | 2 | imaeq2d 6063 | . . . 4 ⊢ (𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌V) = (⦋𝐴 / 𝑥⦌𝐵 “ V)) |
4 | 0ima 6081 | . . . . . 6 ⊢ (∅ “ V) = ∅ | |
5 | 4 | eqcomi 2737 | . . . . 5 ⊢ ∅ = (∅ “ V) |
6 | csbprc 4407 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅) | |
7 | 6 | imaeq1d 6062 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌V) = (∅ “ ⦋𝐴 / 𝑥⦌V)) |
8 | 0ima 6081 | . . . . . 6 ⊢ (∅ “ ⦋𝐴 / 𝑥⦌V) = ∅ | |
9 | 7, 8 | eqtrdi 2784 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌V) = ∅) |
10 | 6 | imaeq1d 6062 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵 “ V) = (∅ “ V)) |
11 | 5, 9, 10 | 3eqtr4a 2794 | . . . 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 2756 | . 2 ⊢ ⦋𝐴 / 𝑥⦌(𝐵 “ V) = (⦋𝐴 / 𝑥⦌𝐵 “ V) |
14 | dfrn4 6206 | . . 3 ⊢ ran 𝐵 = (𝐵 “ V) | |
15 | 14 | csbeq2i 3900 | . 2 ⊢ ⦋𝐴 / 𝑥⦌ran 𝐵 = ⦋𝐴 / 𝑥⦌(𝐵 “ V) |
16 | dfrn4 6206 | . 2 ⊢ ran ⦋𝐴 / 𝑥⦌𝐵 = (⦋𝐴 / 𝑥⦌𝐵 “ V) | |
17 | 13, 15, 16 | 3eqtr4i 2766 | 1 ⊢ ⦋𝐴 / 𝑥⦌ran 𝐵 = ran ⦋𝐴 / 𝑥⦌𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1534 ∈ wcel 2099 Vcvv 3471 ⦋csb 3892 ∅c0 4323 ran crn 5679 “ cima 5681 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-xp 5684 df-rel 5685 df-cnv 5686 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 |
This theorem is referenced by: sbcfg 6720 csbima12gALTVD 44336 |
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