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Mirrors > Home > MPE Home > Th. List > eqfnfv2f | Structured version Visualization version 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 6801 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 6801 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘𝑧))) | |
2 | eqfnfv2f.1 | . . . . 5 ⊢ Ⅎ𝑥𝐹 | |
3 | nfcv 2977 | . . . . 5 ⊢ Ⅎ𝑥𝑧 | |
4 | 2, 3 | nffv 6679 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑧) |
5 | eqfnfv2f.2 | . . . . 5 ⊢ Ⅎ𝑥𝐺 | |
6 | 5, 3 | nffv 6679 | . . . 4 ⊢ Ⅎ𝑥(𝐺‘𝑧) |
7 | 4, 6 | nfeq 2991 | . . 3 ⊢ Ⅎ𝑥(𝐹‘𝑧) = (𝐺‘𝑧) |
8 | nfv 1911 | . . 3 ⊢ Ⅎ𝑧(𝐹‘𝑥) = (𝐺‘𝑥) | |
9 | fveq2 6669 | . . . 4 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥)) | |
10 | fveq2 6669 | . . . 4 ⊢ (𝑧 = 𝑥 → (𝐺‘𝑧) = (𝐺‘𝑥)) | |
11 | 9, 10 | eqeq12d 2837 | . . 3 ⊢ (𝑧 = 𝑥 → ((𝐹‘𝑧) = (𝐺‘𝑧) ↔ (𝐹‘𝑥) = (𝐺‘𝑥))) |
12 | 7, 8, 11 | cbvralw 3441 | . 2 ⊢ (∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘𝑧) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)) |
13 | 1, 12 | syl6bb 289 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 Ⅎwnfc 2961 ∀wral 3138 Fn wfn 6349 ‘cfv 6354 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-fv 6362 |
This theorem is referenced by: aacllem 44901 |
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