Theorem eqfnfv2f 5194

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 5190 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 5190	. 2 ⊢ ((𝐹 Fn A ∧ 𝐺 Fn A) → (𝐹 = 𝐺 ↔ ∀z ∈ A (𝐹‘z) = (𝐺‘z)))
2		eqfnfv2f.1	. . . . 5 ⊢ Ⅎx𝐹
3		nfcv 2160	. . . . 5 ⊢ Ⅎxz
4	2, 3	nffv 5110	. . . 4 ⊢ Ⅎx(𝐹‘z)
5		eqfnfv2f.2	. . . . 5 ⊢ Ⅎx𝐺
6	5, 3	nffv 5110	. . . 4 ⊢ Ⅎx(𝐺‘z)
7	4, 6	nfeq 2167	. . 3 ⊢ Ⅎx(𝐹‘z) = (𝐺‘z)
8		nfv 1402	. . 3 ⊢ Ⅎz(𝐹‘x) = (𝐺‘x)
9		fveq2 5103	. . . 4 ⊢ (z = x → (𝐹‘z) = (𝐹‘x))
10		fveq2 5103	. . . 4 ⊢ (z = x → (𝐺‘z) = (𝐺‘x))
11	9, 10	eqeq12d 2036	. . 3 ⊢ (z = x → ((𝐹‘z) = (𝐺‘z) ↔ (𝐹‘x) = (𝐺‘x)))
12	7, 8, 11	cbvral 2507	. 2 ⊢ (∀z ∈ A (𝐹‘z) = (𝐺‘z) ↔ ∀x ∈ A (𝐹‘x) = (𝐺‘x))
13	1, 12	syl6bb 185	1 ⊢ ((𝐹 Fn A ∧ 𝐺 Fn A) → (𝐹 = 𝐺 ↔ ∀x ∈ A (𝐹‘x) = (𝐺‘x)))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1228  Ⅎwnfc 2147  ∀wral 2284   Fn wfn 4824  ‘cfv 4829

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-csb 2830  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-iota 4794  df-fun 4831  df-fn 4832  df-fv 4837

This theorem is referenced by: (None)