Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > sbcfng | GIF version |
Description: Distribute proper substitution through the function predicate with a domain. (Contributed by Alexander van der Vekens, 15-Jul-2018.) |
Ref | Expression |
---|---|
sbcfng | ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]𝐹 Fn 𝐴 ↔ ⦋𝑋 / 𝑥⦌𝐹 Fn ⦋𝑋 / 𝑥⦌𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fn 4905 | . . . 4 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴)) | |
2 | 1 | a1i 9 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))) |
3 | 2 | sbcbidv 2817 | . 2 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]𝐹 Fn 𝐴 ↔ [𝑋 / 𝑥](Fun 𝐹 ∧ dom 𝐹 = 𝐴))) |
4 | sbcfung 4925 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]Fun 𝐹 ↔ Fun ⦋𝑋 / 𝑥⦌𝐹)) | |
5 | sbceqg 2866 | . . . . 5 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]dom 𝐹 = 𝐴 ↔ ⦋𝑋 / 𝑥⦌dom 𝐹 = ⦋𝑋 / 𝑥⦌𝐴)) | |
6 | csbdmg 4529 | . . . . . 6 ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑥⦌dom 𝐹 = dom ⦋𝑋 / 𝑥⦌𝐹) | |
7 | 6 | eqeq1d 2048 | . . . . 5 ⊢ (𝑋 ∈ 𝑉 → (⦋𝑋 / 𝑥⦌dom 𝐹 = ⦋𝑋 / 𝑥⦌𝐴 ↔ dom ⦋𝑋 / 𝑥⦌𝐹 = ⦋𝑋 / 𝑥⦌𝐴)) |
8 | 5, 7 | bitrd 177 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]dom 𝐹 = 𝐴 ↔ dom ⦋𝑋 / 𝑥⦌𝐹 = ⦋𝑋 / 𝑥⦌𝐴)) |
9 | 4, 8 | anbi12d 442 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (([𝑋 / 𝑥]Fun 𝐹 ∧ [𝑋 / 𝑥]dom 𝐹 = 𝐴) ↔ (Fun ⦋𝑋 / 𝑥⦌𝐹 ∧ dom ⦋𝑋 / 𝑥⦌𝐹 = ⦋𝑋 / 𝑥⦌𝐴))) |
10 | sbcan 2805 | . . 3 ⊢ ([𝑋 / 𝑥](Fun 𝐹 ∧ dom 𝐹 = 𝐴) ↔ ([𝑋 / 𝑥]Fun 𝐹 ∧ [𝑋 / 𝑥]dom 𝐹 = 𝐴)) | |
11 | df-fn 4905 | . . 3 ⊢ (⦋𝑋 / 𝑥⦌𝐹 Fn ⦋𝑋 / 𝑥⦌𝐴 ↔ (Fun ⦋𝑋 / 𝑥⦌𝐹 ∧ dom ⦋𝑋 / 𝑥⦌𝐹 = ⦋𝑋 / 𝑥⦌𝐴)) | |
12 | 9, 10, 11 | 3bitr4g 212 | . 2 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥](Fun 𝐹 ∧ dom 𝐹 = 𝐴) ↔ ⦋𝑋 / 𝑥⦌𝐹 Fn ⦋𝑋 / 𝑥⦌𝐴)) |
13 | 3, 12 | bitrd 177 | 1 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]𝐹 Fn 𝐴 ↔ ⦋𝑋 / 𝑥⦌𝐹 Fn ⦋𝑋 / 𝑥⦌𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1243 ∈ wcel 1393 [wsbc 2764 ⦋csb 2852 dom cdm 4345 Fun wfun 4896 Fn wfn 4897 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-v 2559 df-sbc 2765 df-csb 2853 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-id 4030 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-fun 4904 df-fn 4905 |
This theorem is referenced by: sbcfg 5045 |
Copyright terms: Public domain | W3C validator |