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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 5270 | . . 3 ⊢ (𝐹 = 𝐺 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴)) | |
2 | rneq 4825 | . . . 4 ⊢ (𝐹 = 𝐺 → ran 𝐹 = ran 𝐺) | |
3 | 2 | sseq1d 3166 | . . 3 ⊢ (𝐹 = 𝐺 → (ran 𝐹 ⊆ 𝐵 ↔ ran 𝐺 ⊆ 𝐵)) |
4 | 1, 3 | anbi12d 465 | . 2 ⊢ (𝐹 = 𝐺 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐵))) |
5 | df-f 5186 | . 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | |
6 | df-f 5186 | . 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 1342 ⊆ wss 3111 ran crn 4599 Fn wfn 5177 ⟶wf 5178 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-fun 5184 df-fn 5185 df-f 5186 |
This theorem is referenced by: feq1d 5318 feq1i 5324 f00 5373 f0bi 5374 f0dom0 5375 fconstg 5378 f1eq1 5382 fconst2g 5694 tfrcllemsucfn 6312 tfrcllemsucaccv 6313 tfrcllembxssdm 6315 tfrcllembfn 6316 tfrcllemex 6319 tfrcllemaccex 6320 tfrcllemres 6321 tfrcl 6323 elmapg 6618 ac6sfi 6855 updjud 7038 finomni 7095 exmidomni 7097 mkvprop 7113 1fv 10064 upxp 12813 txcn 12816 dceqnconst 13772 dcapnconst 13773 |
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