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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 5265 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 5265 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘𝑧))) | |
2 | eqfnfv2f.1 | . . . . 5 ⊢ Ⅎ𝑥𝐹 | |
3 | nfcv 2178 | . . . . 5 ⊢ Ⅎ𝑥𝑧 | |
4 | 2, 3 | nffv 5185 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑧) |
5 | eqfnfv2f.2 | . . . . 5 ⊢ Ⅎ𝑥𝐺 | |
6 | 5, 3 | nffv 5185 | . . . 4 ⊢ Ⅎ𝑥(𝐺‘𝑧) |
7 | 4, 6 | nfeq 2185 | . . 3 ⊢ Ⅎ𝑥(𝐹‘𝑧) = (𝐺‘𝑧) |
8 | nfv 1421 | . . 3 ⊢ Ⅎ𝑧(𝐹‘𝑥) = (𝐺‘𝑥) | |
9 | fveq2 5178 | . . . 4 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥)) | |
10 | fveq2 5178 | . . . 4 ⊢ (𝑧 = 𝑥 → (𝐺‘𝑧) = (𝐺‘𝑥)) | |
11 | 9, 10 | eqeq12d 2054 | . . 3 ⊢ (𝑧 = 𝑥 → ((𝐹‘𝑧) = (𝐺‘𝑧) ↔ (𝐹‘𝑥) = (𝐺‘𝑥))) |
12 | 7, 8, 11 | cbvral 2529 | . 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 1243 Ⅎwnfc 2165 ∀wral 2306 Fn wfn 4897 ‘cfv 4902 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-csb 2853 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-iota 4867 df-fun 4904 df-fn 4905 df-fv 4910 |
This theorem is referenced by: (None) |
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