Theorem eqfnfv2f 5290

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 5286 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 29-Jan-2004.)

Hypotheses

Ref	Expression
eqfnfv2f.1	⊢ Ⅎ𝑥𝐹
eqfnfv2f.2	⊢ Ⅎ𝑥𝐺

Assertion

Ref	Expression
eqfnfv2f	⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)))

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝐹(𝑥)   𝐺(𝑥)

Proof of Theorem eqfnfv2f

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqfnfv 5286	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘𝑧)))
2		eqfnfv2f.1	. . . . 5 ⊢ Ⅎ𝑥𝐹
3		nfcv 2219	. . . . 5 ⊢ Ⅎ𝑥𝑧
4	2, 3	nffv 5205	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑧)
5		eqfnfv2f.2	. . . . 5 ⊢ Ⅎ𝑥𝐺
6	5, 3	nffv 5205	. . . 4 ⊢ Ⅎ𝑥(𝐺‘𝑧)
7	4, 6	nfeq 2226	. . 3 ⊢ Ⅎ𝑥(𝐹‘𝑧) = (𝐺‘𝑧)
8		nfv 1461	. . 3 ⊢ Ⅎ𝑧(𝐹‘𝑥) = (𝐺‘𝑥)
9		fveq2 5198	. . . 4 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
10		fveq2 5198	. . . 4 ⊢ (𝑧 = 𝑥 → (𝐺‘𝑧) = (𝐺‘𝑥))
11	9, 10	eqeq12d 2095	. . 3 ⊢ (𝑧 = 𝑥 → ((𝐹‘𝑧) = (𝐺‘𝑧) ↔ (𝐹‘𝑥) = (𝐺‘𝑥)))
12	7, 8, 11	cbvral 2573	. 2 ⊢ (∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘𝑧) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))
13	1, 12	syl6bb 194	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1284  Ⅎwnfc 2206  ∀wral 2348   Fn wfn 4917  ‘cfv 4922

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fn 4925  df-fv 4930

This theorem is referenced by: (None)