Theorem cleqfvf 2881

Description: Equality of functions is determined by their values. Exercise 4 of [TakeutiZaring] p. 28. This version of cleqfv 2880 uses bound variable hypotheses instead of distinct variable conditions.

Hypotheses

Ref	Expression
cleqfvf.1	⊢ (y ∈ F → ∀x y ∈ F)
cleqfvf.2	⊢ (y ∈ G → ∀x y ∈ G)

Assertion

Ref	Expression
cleqfvf	⊢ ((F Fn A ∧ G Fn B) → (F = G ↔ (A = B ∧ ∀x ∈ A (F ‘x) = (G ‘x))))

Distinct variable group(s):   x,y,A   x,B,y   y,F   y,G

Proof of Theorem cleqfvf

Step	Hyp	Ref	Expression
1		cleqfv 2880	. 2 ⊢ ((F Fn A ∧ G Fn B) → (F = G ↔ (A = B ∧ ∀z ∈ A (F ‘z) = (G ‘z))))
2		cleqfvf.1	. . . . . . 7 ⊢ (y ∈ F → ∀x y ∈ F)
3		ax-17 925	. . . . . . 7 ⊢ (y ∈ z → ∀x y ∈ z)
4	2, 3	hbfv 2837	. . . . . 6 ⊢ (y ∈ (F ‘z) → ∀x y ∈ (F ‘z))
5		cleqfvf.2	. . . . . . 7 ⊢ (y ∈ G → ∀x y ∈ G)
6	5, 3	hbfv 2837	. . . . . 6 ⊢ (y ∈ (G ‘z) → ∀x y ∈ (G ‘z))
7	4, 6	hbeq 1171	. . . . 5 ⊢ ((F ‘z) = (G ‘z) → ∀x(F ‘z) = (G ‘z))
8		ax-17 925	. . . . 5 ⊢ ((F ‘x) = (G ‘x) → ∀z(F ‘x) = (G ‘x))
9		fveq2 2832	. . . . . 6 ⊢ (z = x → (F ‘z) = (F ‘x))
10		fveq2 2832	. . . . . 6 ⊢ (z = x → (G ‘z) = (G ‘x))
11	9, 10	cleq12d 1115	. . . . 5 ⊢ (z = x → ((F ‘z) = (G ‘z) ↔ (F ‘x) = (G ‘x)))
12	7, 8, 11	cbvral 1331	. . . 4 ⊢ (∀z ∈ A (F ‘z) = (G ‘z) ↔ ∀x ∈ A (F ‘x) = (G ‘x))
13	12	anbi2i 367	. . 3 ⊢ ((A = B ∧ ∀z ∈ A (F ‘z) = (G ‘z)) ↔ (A = B ∧ ∀x ∈ A (F ‘x) = (G ‘x)))
14	13	bibi2i 460	. 2 ⊢ ((F = G ↔ (A = B ∧ ∀z ∈ A (F ‘z) = (G ‘z))) ↔ (F = G ↔ (A = B ∧ ∀x ∈ A (F ‘x) = (G ‘x))))
15	1, 14	sylib 173	1 ⊢ ((F Fn A ∧ G Fn B) → (F = G ↔ (A = B ∧ ∀x ∈ A (F ‘x) = (G ‘x))))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672   = weq 797   ∈ wel 803   = wceq 1091   ∈ wcel 1092  ∀wral 1201   Fn wfn 2417   ‘cfv 2422

This theorem is referenced by:  pw2en 3348

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438

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