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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 5151 | . . 3 ⊢ (𝐹 = 𝐺 → (Fun 𝐹 ↔ Fun 𝐺)) | |
2 | dmeq 4747 | . . . 4 ⊢ (𝐹 = 𝐺 → dom 𝐹 = dom 𝐺) | |
3 | 2 | eqeq1d 2149 | . . 3 ⊢ (𝐹 = 𝐺 → (dom 𝐹 = 𝐴 ↔ dom 𝐺 = 𝐴)) |
4 | 1, 3 | anbi12d 465 | . 2 ⊢ (𝐹 = 𝐺 → ((Fun 𝐹 ∧ dom 𝐹 = 𝐴) ↔ (Fun 𝐺 ∧ dom 𝐺 = 𝐴))) |
5 | df-fn 5134 | . 2 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴)) | |
6 | df-fn 5134 | . 2 ⊢ (𝐺 Fn 𝐴 ↔ (Fun 𝐺 ∧ dom 𝐺 = 𝐴)) | |
7 | 4, 5, 6 | 3bitr4g 222 | 1 ⊢ (𝐹 = 𝐺 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1332 dom cdm 4547 Fun wfun 5125 Fn wfn 5126 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-opab 3998 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-fun 5133 df-fn 5134 |
This theorem is referenced by: fneq1d 5221 fneq1i 5225 fn0 5250 feq1 5263 foeq1 5349 f1ocnv 5388 mpteqb 5519 eufnfv 5656 tfr0dm 6227 tfrlemiex 6236 tfr1onlemsucfn 6245 tfr1onlemsucaccv 6246 tfr1onlembxssdm 6248 tfr1onlembfn 6249 tfr1onlemex 6252 tfr1onlemaccex 6253 tfr1onlemres 6254 mapval2 6580 elixp2 6604 ixpfn 6606 elixpsn 6637 cc2lem 7098 cc3 7100 |
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