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Mirrors > Home > ILE Home > Th. List > fneq1 | GIF version |
Description: Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.) |
Ref | Expression |
---|---|
fneq1 | ⊢ (𝐹 = 𝐺 → (𝐹 Fn A ↔ 𝐺 Fn A)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funeq 4864 | . . 3 ⊢ (𝐹 = 𝐺 → (Fun 𝐹 ↔ Fun 𝐺)) | |
2 | dmeq 4478 | . . . 4 ⊢ (𝐹 = 𝐺 → dom 𝐹 = dom 𝐺) | |
3 | 2 | eqeq1d 2045 | . . 3 ⊢ (𝐹 = 𝐺 → (dom 𝐹 = A ↔ dom 𝐺 = A)) |
4 | 1, 3 | anbi12d 442 | . 2 ⊢ (𝐹 = 𝐺 → ((Fun 𝐹 ∧ dom 𝐹 = A) ↔ (Fun 𝐺 ∧ dom 𝐺 = A))) |
5 | df-fn 4848 | . 2 ⊢ (𝐹 Fn A ↔ (Fun 𝐹 ∧ dom 𝐹 = A)) | |
6 | df-fn 4848 | . 2 ⊢ (𝐺 Fn A ↔ (Fun 𝐺 ∧ dom 𝐺 = A)) | |
7 | 4, 5, 6 | 3bitr4g 212 | 1 ⊢ (𝐹 = 𝐺 → (𝐹 Fn A ↔ 𝐺 Fn A)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1242 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-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-fun 4847 df-fn 4848 |
This theorem is referenced by: fneq1d 4932 fneq1i 4936 fn0 4961 feq1 4973 foeq1 5045 f1ocnv 5082 mpteqb 5204 eufnfv 5332 tfr0 5878 tfrlemiex 5886 |
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