Theorem eqfnfv3 5210

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 5209	. 2 ⊢ ((𝐹 Fn A ∧ 𝐺 Fn B) → (𝐹 = 𝐺 ↔ (A = B ∧ ∀x ∈ A (𝐹‘x) = (𝐺‘x))))
2		eqss 2954	. . . . 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 431	. . 3 ⊢ ((A = B ∧ ∀x ∈ A (𝐹‘x) = (𝐺‘x)) ↔ ((B ⊆ A ∧ A ⊆ B) ∧ ∀x ∈ A (𝐹‘x) = (𝐺‘x)))
6		anass 381	. . . 4 ⊢ (((B ⊆ A ∧ A ⊆ B) ∧ ∀x ∈ A (𝐹‘x) = (𝐺‘x)) ↔ (B ⊆ A ∧ (A ⊆ B ∧ ∀x ∈ A (𝐹‘x) = (𝐺‘x))))
7		dfss3 2929	. . . . . . 7 ⊢ (A ⊆ B ↔ ∀x ∈ A x ∈ B)
8	7	anbi1i 431	. . . . . 6 ⊢ ((A ⊆ B ∧ ∀x ∈ A (𝐹‘x) = (𝐺‘x)) ↔ (∀x ∈ A x ∈ B ∧ ∀x ∈ A (𝐹‘x) = (𝐺‘x)))
9		r19.26 2435	. . . . . 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 430	. . . 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 1242   ∈ wcel 1390  ∀wral 2300   ⊆ wss 2911   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-bndl 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)