Theorem eqfnfv3 5192

Description: Derive equality of functions from equality of their values. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
eqfnfv3	⊢ ((𝐹 Fn A ∧ 𝐺 Fn B) → (𝐹 = 𝐺 ↔ (B ⊆ A ∧ ∀x ∈ A (x ∈ B ∧ (𝐹‘x) = (𝐺‘x)))))

Distinct variable groups:   x,A   x,𝐹   x,𝐺   x,B

Proof of Theorem eqfnfv3

Step	Hyp	Ref	Expression
1		eqfnfv2 5191	. 2 ⊢ ((𝐹 Fn A ∧ 𝐺 Fn B) → (𝐹 = 𝐺 ↔ (A = B ∧ ∀x ∈ A (𝐹‘x) = (𝐺‘x))))
2		eqss 2937	. . . . 5 ⊢ (A = B ↔ (A ⊆ B ∧ B ⊆ A))
3		ancom 253	. . . . 5 ⊢ ((A ⊆ B ∧ B ⊆ A) ↔ (B ⊆ A ∧ A ⊆ B))
4	2, 3	bitri 173	. . . 4 ⊢ (A = B ↔ (B ⊆ A ∧ A ⊆ B))
5	4	anbi1i 434	. . 3 ⊢ ((A = B ∧ ∀x ∈ A (𝐹‘x) = (𝐺‘x)) ↔ ((B ⊆ A ∧ A ⊆ B) ∧ ∀x ∈ A (𝐹‘x) = (𝐺‘x)))
6		anass 383	. . . 4 ⊢ (((B ⊆ A ∧ A ⊆ B) ∧ ∀x ∈ A (𝐹‘x) = (𝐺‘x)) ↔ (B ⊆ A ∧ (A ⊆ B ∧ ∀x ∈ A (𝐹‘x) = (𝐺‘x))))
7		dfss3 2912	. . . . . . 7 ⊢ (A ⊆ B ↔ ∀x ∈ A x ∈ B)
8	7	anbi1i 434	. . . . . 6 ⊢ ((A ⊆ B ∧ ∀x ∈ A (𝐹‘x) = (𝐺‘x)) ↔ (∀x ∈ A x ∈ B ∧ ∀x ∈ A (𝐹‘x) = (𝐺‘x)))
9		r19.26 2419	. . . . . 6 ⊢ (∀x ∈ A (x ∈ B ∧ (𝐹‘x) = (𝐺‘x)) ↔ (∀x ∈ A x ∈ B ∧ ∀x ∈ A (𝐹‘x) = (𝐺‘x)))
10	8, 9	bitr4i 176	. . . . 5 ⊢ ((A ⊆ B ∧ ∀x ∈ A (𝐹‘x) = (𝐺‘x)) ↔ ∀x ∈ A (x ∈ B ∧ (𝐹‘x) = (𝐺‘x)))
11	10	anbi2i 433	. . . 4 ⊢ ((B ⊆ A ∧ (A ⊆ B ∧ ∀x ∈ A (𝐹‘x) = (𝐺‘x))) ↔ (B ⊆ A ∧ ∀x ∈ A (x ∈ B ∧ (𝐹‘x) = (𝐺‘x))))
12	6, 11	bitri 173	. . 3 ⊢ (((B ⊆ A ∧ A ⊆ B) ∧ ∀x ∈ A (𝐹‘x) = (𝐺‘x)) ↔ (B ⊆ A ∧ ∀x ∈ A (x ∈ B ∧ (𝐹‘x) = (𝐺‘x))))
13	5, 12	bitri 173	. 2 ⊢ ((A = B ∧ ∀x ∈ A (𝐹‘x) = (𝐺‘x)) ↔ (B ⊆ A ∧ ∀x ∈ A (x ∈ B ∧ (𝐹‘x) = (𝐺‘x))))
14	1, 13	syl6bb 185	1 ⊢ ((𝐹 Fn A ∧ 𝐺 Fn B) → (𝐹 = 𝐺 ↔ (B ⊆ A ∧ ∀x ∈ A (x ∈ B ∧ (𝐹‘x) = (𝐺‘x)))))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1228   ∈ wcel 1374  ∀wral 2284   ⊆ wss 2894   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)