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Mirrors > Home > NFE Home > Th. List > csbresg | GIF version |
Description: Distribute proper substitution through the restriction of a class. csbresg 4976 is derived from the virtual deduction proof csbresgVD in set.mm. (Contributed by Alan Sare, 10-Nov-2012.) |
Ref | Expression |
---|---|
csbresg | ⊢ (A ∈ V → [A / x](B ↾ C) = ([A / x]B ↾ [A / x]C)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbing 3462 | . . 3 ⊢ (A ∈ V → [A / x](B ∩ (C × V)) = ([A / x]B ∩ [A / x](C × V))) | |
2 | csbxpg 4813 | . . . . 5 ⊢ (A ∈ V → [A / x](C × V) = ([A / x]C × [A / x]V)) | |
3 | csbconstg 3150 | . . . . . 6 ⊢ (A ∈ V → [A / x]V = V) | |
4 | 3 | xpeq2d 4808 | . . . . 5 ⊢ (A ∈ V → ([A / x]C × [A / x]V) = ([A / x]C × V)) |
5 | 2, 4 | eqtrd 2385 | . . . 4 ⊢ (A ∈ V → [A / x](C × V) = ([A / x]C × V)) |
6 | 5 | ineq2d 3457 | . . 3 ⊢ (A ∈ V → ([A / x]B ∩ [A / x](C × V)) = ([A / x]B ∩ ([A / x]C × V))) |
7 | 1, 6 | eqtrd 2385 | . 2 ⊢ (A ∈ V → [A / x](B ∩ (C × V)) = ([A / x]B ∩ ([A / x]C × V))) |
8 | df-res 4788 | . . 3 ⊢ (B ↾ C) = (B ∩ (C × V)) | |
9 | 8 | csbeq2i 3162 | . 2 ⊢ [A / x](B ↾ C) = [A / x](B ∩ (C × V)) |
10 | df-res 4788 | . 2 ⊢ ([A / x]B ↾ [A / x]C) = ([A / x]B ∩ ([A / x]C × V)) | |
11 | 7, 9, 10 | 3eqtr4g 2410 | 1 ⊢ (A ∈ V → [A / x](B ↾ C) = ([A / x]B ↾ [A / x]C)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1642 ∈ wcel 1710 Vcvv 2859 [csb 3136 ∩ cin 3208 × cxp 4770 ↾ cres 4774 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-sbc 3047 df-csb 3137 df-nin 3211 df-compl 3212 df-in 3213 df-opab 4623 df-xp 4784 df-res 4788 |
This theorem is referenced by: (None) |
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