Theorem eqfnfv2f 7010

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 7006 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 7006	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘𝑧)))
2		eqfnfv2f.1	. . . . 5 ⊢ Ⅎ𝑥𝐹
3		nfcv 2892	. . . . 5 ⊢ Ⅎ𝑥𝑧
4	2, 3	nffv 6871	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑧)
5		eqfnfv2f.2	. . . . 5 ⊢ Ⅎ𝑥𝐺
6	5, 3	nffv 6871	. . . 4 ⊢ Ⅎ𝑥(𝐺‘𝑧)
7	4, 6	nfeq 2906	. . 3 ⊢ Ⅎ𝑥(𝐹‘𝑧) = (𝐺‘𝑧)
8		nfv 1914	. . 3 ⊢ Ⅎ𝑧(𝐹‘𝑥) = (𝐺‘𝑥)
9		fveq2 6861	. . . 4 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
10		fveq2 6861	. . . 4 ⊢ (𝑧 = 𝑥 → (𝐺‘𝑧) = (𝐺‘𝑥))
11	9, 10	eqeq12d 2746	. . 3 ⊢ (𝑧 = 𝑥 → ((𝐹‘𝑧) = (𝐺‘𝑧) ↔ (𝐹‘𝑥) = (𝐺‘𝑥)))
12	7, 8, 11	cbvralw 3282	. 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 1540  Ⅎwnfc 2877  ∀wral 3045   Fn wfn 6509  ‘cfv 6514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-fv 6522

This theorem is referenced by:  aacllem  49794