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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 5781 | . . 3 ⊢ ⦋𝐴 / 𝑥⦌(𝐵 “ V) = (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌V) | |
2 | csbconstg 3795 | . . . . 5 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌V = V) | |
3 | 2 | imaeq2d 5764 | . . . 4 ⊢ (𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌V) = (⦋𝐴 / 𝑥⦌𝐵 “ V)) |
4 | 0ima 5780 | . . . . . 6 ⊢ (∅ “ V) = ∅ | |
5 | 4 | eqcomi 2781 | . . . . 5 ⊢ ∅ = (∅ “ V) |
6 | csbprc 4238 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅) | |
7 | 6 | imaeq1d 5763 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌V) = (∅ “ ⦋𝐴 / 𝑥⦌V)) |
8 | 0ima 5780 | . . . . . 6 ⊢ (∅ “ ⦋𝐴 / 𝑥⦌V) = ∅ | |
9 | 7, 8 | syl6eq 2824 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌V) = ∅) |
10 | 6 | imaeq1d 5763 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵 “ V) = (∅ “ V)) |
11 | 5, 9, 10 | 3eqtr4a 2834 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌V) = (⦋𝐴 / 𝑥⦌𝐵 “ V)) |
12 | 3, 11 | pm2.61i 177 | . . 3 ⊢ (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌V) = (⦋𝐴 / 𝑥⦌𝐵 “ V) |
13 | 1, 12 | eqtri 2796 | . 2 ⊢ ⦋𝐴 / 𝑥⦌(𝐵 “ V) = (⦋𝐴 / 𝑥⦌𝐵 “ V) |
14 | dfrn4 5892 | . . 3 ⊢ ran 𝐵 = (𝐵 “ V) | |
15 | 14 | csbeq2i 4251 | . 2 ⊢ ⦋𝐴 / 𝑥⦌ran 𝐵 = ⦋𝐴 / 𝑥⦌(𝐵 “ V) |
16 | dfrn4 5892 | . 2 ⊢ ran ⦋𝐴 / 𝑥⦌𝐵 = (⦋𝐴 / 𝑥⦌𝐵 “ V) | |
17 | 13, 15, 16 | 3eqtr4i 2806 | 1 ⊢ ⦋𝐴 / 𝑥⦌ran 𝐵 = ran ⦋𝐴 / 𝑥⦌𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1507 ∈ wcel 2048 Vcvv 3409 ⦋csb 3782 ∅c0 4173 ran crn 5401 “ cima 5403 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-sep 5054 ax-nul 5061 ax-pr 5180 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-br 4924 df-opab 4986 df-xp 5406 df-rel 5407 df-cnv 5408 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 |
This theorem is referenced by: sbcfg 6336 csbima12gALTVD 40594 |
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