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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 4848 | . . . 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 2811 | . 2 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / x]𝐹 Fn A ↔ [𝑋 / x](Fun 𝐹 ∧ dom 𝐹 = A))) |
4 | sbcfung 4868 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / x]Fun 𝐹 ↔ Fun ⦋𝑋 / x⦌𝐹)) | |
5 | sbceqg 2860 | . . . . 5 ⊢ (𝑋 ∈ 𝑉 → ([𝑋 / x]dom 𝐹 = A ↔ ⦋𝑋 / x⦌dom 𝐹 = ⦋𝑋 / x⦌A)) | |
6 | csbdmg 4472 | . . . . . 6 ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / x⦌dom 𝐹 = dom ⦋𝑋 / x⦌𝐹) | |
7 | 6 | eqeq1d 2045 | . . . . 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 2799 | . . 3 ⊢ ([𝑋 / x](Fun 𝐹 ∧ dom 𝐹 = A) ↔ ([𝑋 / x]Fun 𝐹 ∧ [𝑋 / x]dom 𝐹 = A)) | |
11 | df-fn 4848 | . . 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 1242 ∈ wcel 1390 [wsbc 2758 ⦋csb 2846 dom cdm 4288 Fun wfun 4839 Fn wfn 4840 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-v 2553 df-sbc 2759 df-csb 2847 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-id 4021 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-fun 4847 df-fn 4848 |
This theorem is referenced by: sbcfg 4988 |
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