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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 5593 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 5593 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘𝑧))) | |
2 | eqfnfv2f.1 | . . . . 5 ⊢ Ⅎ𝑥𝐹 | |
3 | nfcv 2312 | . . . . 5 ⊢ Ⅎ𝑥𝑧 | |
4 | 2, 3 | nffv 5506 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑧) |
5 | eqfnfv2f.2 | . . . . 5 ⊢ Ⅎ𝑥𝐺 | |
6 | 5, 3 | nffv 5506 | . . . 4 ⊢ Ⅎ𝑥(𝐺‘𝑧) |
7 | 4, 6 | nfeq 2320 | . . 3 ⊢ Ⅎ𝑥(𝐹‘𝑧) = (𝐺‘𝑧) |
8 | nfv 1521 | . . 3 ⊢ Ⅎ𝑧(𝐹‘𝑥) = (𝐺‘𝑥) | |
9 | fveq2 5496 | . . . 4 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥)) | |
10 | fveq2 5496 | . . . 4 ⊢ (𝑧 = 𝑥 → (𝐺‘𝑧) = (𝐺‘𝑥)) | |
11 | 9, 10 | eqeq12d 2185 | . . 3 ⊢ (𝑧 = 𝑥 → ((𝐹‘𝑧) = (𝐺‘𝑧) ↔ (𝐹‘𝑥) = (𝐺‘𝑥))) |
12 | 7, 8, 11 | cbvral 2692 | . 2 ⊢ (∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘𝑧) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)) |
13 | 1, 12 | bitrdi 195 | 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 103 ↔ wb 104 = wceq 1348 Ⅎwnfc 2299 ∀wral 2448 Fn wfn 5193 ‘cfv 5198 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-csb 3050 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fn 5201 df-fv 5206 |
This theorem is referenced by: (None) |
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