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Mirrors > Home > ILE Home > Th. List > fveq12d | GIF version |
Description: Equality deduction for function value. (Contributed by FL, 22-Dec-2008.) |
Ref | Expression |
---|---|
fveq12d.1 | ⊢ (𝜑 → 𝐹 = 𝐺) |
fveq12d.2 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
fveq12d | ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq12d.1 | . . 3 ⊢ (𝜑 → 𝐹 = 𝐺) | |
2 | 1 | fveq1d 5377 | . 2 ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐴)) |
3 | fveq12d.2 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
4 | 3 | fveq2d 5379 | . 2 ⊢ (𝜑 → (𝐺‘𝐴) = (𝐺‘𝐵)) |
5 | 2, 4 | eqtrd 2147 | 1 ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1314 ‘cfv 5081 |
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 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-rex 2396 df-v 2659 df-un 3041 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-br 3896 df-iota 5046 df-fv 5089 |
This theorem is referenced by: nffvd 5387 fvsng 5570 tfrlem3ag 6160 tfrlem3a 6161 tfrlemi1 6183 tfr1onlem3ag 6188 omp1eomlem 6931 seq3shft 10503 climshft2 10967 fsum3 11048 ctiunctlemfo 11795 reldvg 12603 dvfvalap 12605 |
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