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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 6035 | . . 3 ⊢ ⦋𝐴 / 𝑥⦌(𝐵 “ V) = (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌V) | |
2 | csbconstg 3878 | . . . . 5 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌V = V) | |
3 | 2 | imaeq2d 6017 | . . . 4 ⊢ (𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌V) = (⦋𝐴 / 𝑥⦌𝐵 “ V)) |
4 | 0ima 6034 | . . . . . 6 ⊢ (∅ “ V) = ∅ | |
5 | 4 | eqcomi 2742 | . . . . 5 ⊢ ∅ = (∅ “ V) |
6 | csbprc 4370 | . . . . . . 7 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅) | |
7 | 6 | imaeq1d 6016 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌V) = (∅ “ ⦋𝐴 / 𝑥⦌V)) |
8 | 0ima 6034 | . . . . . 6 ⊢ (∅ “ ⦋𝐴 / 𝑥⦌V) = ∅ | |
9 | 7, 8 | eqtrdi 2789 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌V) = ∅) |
10 | 6 | imaeq1d 6016 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵 “ V) = (∅ “ V)) |
11 | 5, 9, 10 | 3eqtr4a 2799 | . . . 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 2761 | . 2 ⊢ ⦋𝐴 / 𝑥⦌(𝐵 “ V) = (⦋𝐴 / 𝑥⦌𝐵 “ V) |
14 | dfrn4 6158 | . . 3 ⊢ ran 𝐵 = (𝐵 “ V) | |
15 | 14 | csbeq2i 3867 | . 2 ⊢ ⦋𝐴 / 𝑥⦌ran 𝐵 = ⦋𝐴 / 𝑥⦌(𝐵 “ V) |
16 | dfrn4 6158 | . 2 ⊢ ran ⦋𝐴 / 𝑥⦌𝐵 = (⦋𝐴 / 𝑥⦌𝐵 “ V) | |
17 | 13, 15, 16 | 3eqtr4i 2771 | 1 ⊢ ⦋𝐴 / 𝑥⦌ran 𝐵 = ran ⦋𝐴 / 𝑥⦌𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1542 ∈ wcel 2107 Vcvv 3447 ⦋csb 3859 ∅c0 4286 ran crn 5638 “ cima 5640 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 df-opab 5172 df-xp 5643 df-rel 5644 df-cnv 5645 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 |
This theorem is referenced by: sbcfg 6670 csbima12gALTVD 43271 |
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