Theorem eqfnfv2f 5212

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 5208 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 5208	. 2 ⊢ ((𝐹 Fn A ∧ 𝐺 Fn A) → (𝐹 = 𝐺 ↔ ∀z ∈ A (𝐹‘z) = (𝐺‘z)))
2		eqfnfv2f.1	. . . . 5 ⊢ Ⅎx𝐹
3		nfcv 2175	. . . . 5 ⊢ Ⅎxz
4	2, 3	nffv 5128	. . . 4 ⊢ Ⅎx(𝐹‘z)
5		eqfnfv2f.2	. . . . 5 ⊢ Ⅎx𝐺
6	5, 3	nffv 5128	. . . 4 ⊢ Ⅎx(𝐺‘z)
7	4, 6	nfeq 2182	. . 3 ⊢ Ⅎx(𝐹‘z) = (𝐺‘z)
8		nfv 1418	. . 3 ⊢ Ⅎz(𝐹‘x) = (𝐺‘x)
9		fveq2 5121	. . . 4 ⊢ (z = x → (𝐹‘z) = (𝐹‘x))
10		fveq2 5121	. . . 4 ⊢ (z = x → (𝐺‘z) = (𝐺‘x))
11	9, 10	eqeq12d 2051	. . 3 ⊢ (z = x → ((𝐹‘z) = (𝐺‘z) ↔ (𝐹‘x) = (𝐺‘x)))
12	7, 8, 11	cbvral 2523	. 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 1242  Ⅎwnfc 2162  ∀wral 2300   Fn wfn 4840  ‘cfv 4845

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853

This theorem is referenced by: (None)