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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 5284 | . . 3 ⊢ (𝐹 = 𝐺 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴)) | |
2 | rneq 4836 | . . . 4 ⊢ (𝐹 = 𝐺 → ran 𝐹 = ran 𝐺) | |
3 | 2 | sseq1d 3176 | . . 3 ⊢ (𝐹 = 𝐺 → (ran 𝐹 ⊆ 𝐵 ↔ ran 𝐺 ⊆ 𝐵)) |
4 | 1, 3 | anbi12d 470 | . 2 ⊢ (𝐹 = 𝐺 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐵))) |
5 | df-f 5200 | . 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | |
6 | df-f 5200 | . 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 1348 ⊆ wss 3121 ran crn 4610 Fn wfn 5191 ⟶wf 5192 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-sn 3587 df-pr 3588 df-op 3590 df-br 3988 df-opab 4049 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-fun 5198 df-fn 5199 df-f 5200 |
This theorem is referenced by: feq1d 5332 feq1i 5338 f00 5387 f0bi 5388 f0dom0 5389 fconstg 5392 f1eq1 5396 fconst2g 5708 tfrcllemsucfn 6329 tfrcllemsucaccv 6330 tfrcllembxssdm 6332 tfrcllembfn 6333 tfrcllemex 6336 tfrcllemaccex 6337 tfrcllemres 6338 tfrcl 6340 elmapg 6635 ac6sfi 6872 updjud 7055 finomni 7112 exmidomni 7114 mkvprop 7130 1fv 10082 upxp 13025 txcn 13028 dceqnconst 14051 dcapnconst 14052 |
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