Theorem eqfnfv2f 6968

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 6964 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 6964	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘𝑧)))
2		eqfnfv2f.1	. . . . 5 ⊢ Ⅎ𝑥𝐹
3		nfcv 2894	. . . . 5 ⊢ Ⅎ𝑥𝑧
4	2, 3	nffv 6832	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑧)
5		eqfnfv2f.2	. . . . 5 ⊢ Ⅎ𝑥𝐺
6	5, 3	nffv 6832	. . . 4 ⊢ Ⅎ𝑥(𝐺‘𝑧)
7	4, 6	nfeq 2908	. . 3 ⊢ Ⅎ𝑥(𝐹‘𝑧) = (𝐺‘𝑧)
8		nfv 1915	. . 3 ⊢ Ⅎ𝑧(𝐹‘𝑥) = (𝐺‘𝑥)
9		fveq2 6822	. . . 4 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
10		fveq2 6822	. . . 4 ⊢ (𝑧 = 𝑥 → (𝐺‘𝑧) = (𝐺‘𝑥))
11	9, 10	eqeq12d 2747	. . 3 ⊢ (𝑧 = 𝑥 → ((𝐹‘𝑧) = (𝐺‘𝑧) ↔ (𝐹‘𝑥) = (𝐺‘𝑥)))
12	7, 8, 11	cbvralw 3274	. 2 ⊢ (∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘𝑧) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))
13	1, 12	bitrdi 287	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  Ⅎwnfc 2879  ∀wral 3047   Fn wfn 6476  ‘cfv 6481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-fv 6489

This theorem is referenced by:  aacllem  49839