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Mirrors > Home > ILE Home > Th. List > oveq | GIF version |
Description: Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.) |
Ref | Expression |
---|---|
oveq | ⊢ (𝐹 = 𝐺 → (𝐴𝐹𝐵) = (𝐴𝐺𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq1 5388 | . 2 ⊢ (𝐹 = 𝐺 → (𝐹‘〈𝐴, 𝐵〉) = (𝐺‘〈𝐴, 𝐵〉)) | |
2 | df-ov 5745 | . 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
3 | df-ov 5745 | . 2 ⊢ (𝐴𝐺𝐵) = (𝐺‘〈𝐴, 𝐵〉) | |
4 | 1, 2, 3 | 3eqtr4g 2175 | 1 ⊢ (𝐹 = 𝐺 → (𝐴𝐹𝐵) = (𝐴𝐺𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1316 〈cop 3500 ‘cfv 5093 (class class class)co 5742 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-rex 2399 df-uni 3707 df-br 3900 df-iota 5058 df-fv 5101 df-ov 5745 |
This theorem is referenced by: oveqi 5755 oveqd 5759 ovmpodf 5870 ovmpodv2 5872 mapxpen 6710 ispsmet 12419 ismet 12440 isxmet 12441 |
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