![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > feq1 | GIF version |
Description: Equality theorem for functions. (Contributed by NM, 1-Aug-1994.) |
Ref | Expression |
---|---|
feq1 | ⊢ (𝐹 = 𝐺 → (𝐹:𝐴⟶𝐵 ↔ 𝐺:𝐴⟶𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fneq1 5147 | . . 3 ⊢ (𝐹 = 𝐺 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴)) | |
2 | rneq 4704 | . . . 4 ⊢ (𝐹 = 𝐺 → ran 𝐹 = ran 𝐺) | |
3 | 2 | sseq1d 3076 | . . 3 ⊢ (𝐹 = 𝐺 → (ran 𝐹 ⊆ 𝐵 ↔ ran 𝐺 ⊆ 𝐵)) |
4 | 1, 3 | anbi12d 460 | . 2 ⊢ (𝐹 = 𝐺 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐵))) |
5 | df-f 5063 | . 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | |
6 | df-f 5063 | . 2 ⊢ (𝐺:𝐴⟶𝐵 ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐵)) | |
7 | 4, 5, 6 | 3bitr4g 222 | 1 ⊢ (𝐹 = 𝐺 → (𝐹:𝐴⟶𝐵 ↔ 𝐺:𝐴⟶𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1299 ⊆ wss 3021 ran crn 4478 Fn wfn 5054 ⟶wf 5055 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-v 2643 df-un 3025 df-in 3027 df-ss 3034 df-sn 3480 df-pr 3481 df-op 3483 df-br 3876 df-opab 3930 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-fun 5061 df-fn 5062 df-f 5063 |
This theorem is referenced by: feq1d 5195 feq1i 5201 f00 5250 f0bi 5251 f0dom0 5252 fconstg 5255 f1eq1 5259 fconst2g 5567 tfrcllemsucfn 6180 tfrcllemsucaccv 6181 tfrcllembxssdm 6183 tfrcllembfn 6184 tfrcllemex 6187 tfrcllemaccex 6188 tfrcllemres 6189 tfrcl 6191 elmapg 6485 ac6sfi 6721 updjud 6882 finomni 6924 exmidomni 6926 mkvprop 6943 1fv 9757 upxp 12222 txcn 12225 |
Copyright terms: Public domain | W3C validator |