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Mirrors > Home > ILE Home > Th. List > eqfnfv2f | GIF version |
Description: Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). This version of eqfnfv 5208 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 29-Jan-2004.) |
Ref | Expression |
---|---|
eqfnfv2f.1 | ⊢ Ⅎx𝐹 |
eqfnfv2f.2 | ⊢ Ⅎx𝐺 |
Ref | Expression |
---|---|
eqfnfv2f | ⊢ ((𝐹 Fn A ∧ 𝐺 Fn A) → (𝐹 = 𝐺 ↔ ∀x ∈ A (𝐹‘x) = (𝐺‘x))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqfnfv 5208 | . 2 ⊢ ((𝐹 Fn A ∧ 𝐺 Fn A) → (𝐹 = 𝐺 ↔ ∀z ∈ A (𝐹‘z) = (𝐺‘z))) | |
2 | eqfnfv2f.1 | . . . . 5 ⊢ Ⅎx𝐹 | |
3 | nfcv 2175 | . . . . 5 ⊢ Ⅎxz | |
4 | 2, 3 | nffv 5128 | . . . 4 ⊢ Ⅎx(𝐹‘z) |
5 | eqfnfv2f.2 | . . . . 5 ⊢ Ⅎx𝐺 | |
6 | 5, 3 | nffv 5128 | . . . 4 ⊢ Ⅎx(𝐺‘z) |
7 | 4, 6 | nfeq 2182 | . . 3 ⊢ Ⅎx(𝐹‘z) = (𝐺‘z) |
8 | nfv 1418 | . . 3 ⊢ Ⅎz(𝐹‘x) = (𝐺‘x) | |
9 | fveq2 5121 | . . . 4 ⊢ (z = x → (𝐹‘z) = (𝐹‘x)) | |
10 | fveq2 5121 | . . . 4 ⊢ (z = x → (𝐺‘z) = (𝐺‘x)) | |
11 | 9, 10 | eqeq12d 2051 | . . 3 ⊢ (z = x → ((𝐹‘z) = (𝐺‘z) ↔ (𝐹‘x) = (𝐺‘x))) |
12 | 7, 8, 11 | cbvral 2523 | . 2 ⊢ (∀z ∈ A (𝐹‘z) = (𝐺‘z) ↔ ∀x ∈ A (𝐹‘x) = (𝐺‘x)) |
13 | 1, 12 | syl6bb 185 | 1 ⊢ ((𝐹 Fn A ∧ 𝐺 Fn A) → (𝐹 = 𝐺 ↔ ∀x ∈ A (𝐹‘x) = (𝐺‘x))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1242 Ⅎwnfc 2162 ∀wral 2300 Fn wfn 4840 ‘cfv 4845 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-sbc 2759 df-csb 2847 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-iota 4810 df-fun 4847 df-fn 4848 df-fv 4853 |
This theorem is referenced by: (None) |
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