Theorem eqfnfv2f 5659

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 5655 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 5655	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘𝑧)))
2		eqfnfv2f.1	. . . . 5 ⊢ Ⅎ𝑥𝐹
3		nfcv 2336	. . . . 5 ⊢ Ⅎ𝑥𝑧
4	2, 3	nffv 5564	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑧)
5		eqfnfv2f.2	. . . . 5 ⊢ Ⅎ𝑥𝐺
6	5, 3	nffv 5564	. . . 4 ⊢ Ⅎ𝑥(𝐺‘𝑧)
7	4, 6	nfeq 2344	. . 3 ⊢ Ⅎ𝑥(𝐹‘𝑧) = (𝐺‘𝑧)
8		nfv 1539	. . 3 ⊢ Ⅎ𝑧(𝐹‘𝑥) = (𝐺‘𝑥)
9		fveq2 5554	. . . 4 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
10		fveq2 5554	. . . 4 ⊢ (𝑧 = 𝑥 → (𝐺‘𝑧) = (𝐺‘𝑥))
11	9, 10	eqeq12d 2208	. . 3 ⊢ (𝑧 = 𝑥 → ((𝐹‘𝑧) = (𝐺‘𝑧) ↔ (𝐹‘𝑥) = (𝐺‘𝑥)))
12	7, 8, 11	cbvral 2722	. 2 ⊢ (∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘𝑧) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))
13	1, 12	bitrdi 196	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364  Ⅎwnfc 2323  ∀wral 2472   Fn wfn 5249  ‘cfv 5254

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262

This theorem is referenced by: (None)