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Mirrors > Home > ILE Home > Th. List > eqfnfv3 | GIF version |
Description: Derive equality of functions from equality of their values. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
eqfnfv3 | ⊢ ((𝐹 Fn A ∧ 𝐺 Fn B) → (𝐹 = 𝐺 ↔ (B ⊆ A ∧ ∀x ∈ A (x ∈ B ∧ (𝐹‘x) = (𝐺‘x))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqfnfv2 5209 | . 2 ⊢ ((𝐹 Fn A ∧ 𝐺 Fn B) → (𝐹 = 𝐺 ↔ (A = B ∧ ∀x ∈ A (𝐹‘x) = (𝐺‘x)))) | |
2 | eqss 2954 | . . . . 5 ⊢ (A = B ↔ (A ⊆ B ∧ B ⊆ A)) | |
3 | ancom 253 | . . . . 5 ⊢ ((A ⊆ B ∧ B ⊆ A) ↔ (B ⊆ A ∧ A ⊆ B)) | |
4 | 2, 3 | bitri 173 | . . . 4 ⊢ (A = B ↔ (B ⊆ A ∧ A ⊆ B)) |
5 | 4 | anbi1i 431 | . . 3 ⊢ ((A = B ∧ ∀x ∈ A (𝐹‘x) = (𝐺‘x)) ↔ ((B ⊆ A ∧ A ⊆ B) ∧ ∀x ∈ A (𝐹‘x) = (𝐺‘x))) |
6 | anass 381 | . . . 4 ⊢ (((B ⊆ A ∧ A ⊆ B) ∧ ∀x ∈ A (𝐹‘x) = (𝐺‘x)) ↔ (B ⊆ A ∧ (A ⊆ B ∧ ∀x ∈ A (𝐹‘x) = (𝐺‘x)))) | |
7 | dfss3 2929 | . . . . . . 7 ⊢ (A ⊆ B ↔ ∀x ∈ A x ∈ B) | |
8 | 7 | anbi1i 431 | . . . . . 6 ⊢ ((A ⊆ B ∧ ∀x ∈ A (𝐹‘x) = (𝐺‘x)) ↔ (∀x ∈ A x ∈ B ∧ ∀x ∈ A (𝐹‘x) = (𝐺‘x))) |
9 | r19.26 2435 | . . . . . 6 ⊢ (∀x ∈ A (x ∈ B ∧ (𝐹‘x) = (𝐺‘x)) ↔ (∀x ∈ A x ∈ B ∧ ∀x ∈ A (𝐹‘x) = (𝐺‘x))) | |
10 | 8, 9 | bitr4i 176 | . . . . 5 ⊢ ((A ⊆ B ∧ ∀x ∈ A (𝐹‘x) = (𝐺‘x)) ↔ ∀x ∈ A (x ∈ B ∧ (𝐹‘x) = (𝐺‘x))) |
11 | 10 | anbi2i 430 | . . . 4 ⊢ ((B ⊆ A ∧ (A ⊆ B ∧ ∀x ∈ A (𝐹‘x) = (𝐺‘x))) ↔ (B ⊆ A ∧ ∀x ∈ A (x ∈ B ∧ (𝐹‘x) = (𝐺‘x)))) |
12 | 6, 11 | bitri 173 | . . 3 ⊢ (((B ⊆ A ∧ A ⊆ B) ∧ ∀x ∈ A (𝐹‘x) = (𝐺‘x)) ↔ (B ⊆ A ∧ ∀x ∈ A (x ∈ B ∧ (𝐹‘x) = (𝐺‘x)))) |
13 | 5, 12 | bitri 173 | . 2 ⊢ ((A = B ∧ ∀x ∈ A (𝐹‘x) = (𝐺‘x)) ↔ (B ⊆ A ∧ ∀x ∈ A (x ∈ B ∧ (𝐹‘x) = (𝐺‘x)))) |
14 | 1, 13 | syl6bb 185 | 1 ⊢ ((𝐹 Fn A ∧ 𝐺 Fn B) → (𝐹 = 𝐺 ↔ (B ⊆ A ∧ ∀x ∈ A (x ∈ B ∧ (𝐹‘x) = (𝐺‘x))))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1242 ∈ wcel 1390 ∀wral 2300 ⊆ wss 2911 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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