Theorem eqfnov 5549

Description: Equality of two operations is determined by their values. (Contributed by NM, 1-Sep-2005.)

Assertion

Ref	Expression
eqfnov	⊢ ((𝐹 Fn (A × B) ∧ 𝐺 Fn (𝐶 × 𝐷)) → (𝐹 = 𝐺 ↔ ((A × B) = (𝐶 × 𝐷) ∧ ∀x ∈ A ∀y ∈ B (x𝐹y) = (x𝐺y))))

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

Allowed substitution hints:   𝐶(x,y)   𝐷(x,y)

Proof of Theorem eqfnov

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqfnfv2 5209	. 2 ⊢ ((𝐹 Fn (A × B) ∧ 𝐺 Fn (𝐶 × 𝐷)) → (𝐹 = 𝐺 ↔ ((A × B) = (𝐶 × 𝐷) ∧ ∀z ∈ (A × B)(𝐹‘z) = (𝐺‘z))))
2		fveq2 5121	. . . . . 6 ⊢ (z = ⟨x, y⟩ → (𝐹‘z) = (𝐹‘⟨x, y⟩))
3		fveq2 5121	. . . . . 6 ⊢ (z = ⟨x, y⟩ → (𝐺‘z) = (𝐺‘⟨x, y⟩))
4	2, 3	eqeq12d 2051	. . . . 5 ⊢ (z = ⟨x, y⟩ → ((𝐹‘z) = (𝐺‘z) ↔ (𝐹‘⟨x, y⟩) = (𝐺‘⟨x, y⟩)))
5		df-ov 5458	. . . . . 6 ⊢ (x𝐹y) = (𝐹‘⟨x, y⟩)
6		df-ov 5458	. . . . . 6 ⊢ (x𝐺y) = (𝐺‘⟨x, y⟩)
7	5, 6	eqeq12i 2050	. . . . 5 ⊢ ((x𝐹y) = (x𝐺y) ↔ (𝐹‘⟨x, y⟩) = (𝐺‘⟨x, y⟩))
8	4, 7	syl6bbr 187	. . . 4 ⊢ (z = ⟨x, y⟩ → ((𝐹‘z) = (𝐺‘z) ↔ (x𝐹y) = (x𝐺y)))
9	8	ralxp 4422	. . 3 ⊢ (∀z ∈ (A × B)(𝐹‘z) = (𝐺‘z) ↔ ∀x ∈ A ∀y ∈ B (x𝐹y) = (x𝐺y))
10	9	anbi2i 430	. 2 ⊢ (((A × B) = (𝐶 × 𝐷) ∧ ∀z ∈ (A × B)(𝐹‘z) = (𝐺‘z)) ↔ ((A × B) = (𝐶 × 𝐷) ∧ ∀x ∈ A ∀y ∈ B (x𝐹y) = (x𝐺y)))
11	1, 10	syl6bb 185	1 ⊢ ((𝐹 Fn (A × B) ∧ 𝐺 Fn (𝐶 × 𝐷)) → (𝐹 = 𝐺 ↔ ((A × B) = (𝐶 × 𝐷) ∧ ∀x ∈ A ∀y ∈ B (x𝐹y) = (x𝐺y))))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242  ∀wral 2300  ⟨cop 3370   × cxp 4286   Fn wfn 4840  ‘cfv 4845  (class class class)co 5455

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-iun 3650  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  df-ov 5458

This theorem is referenced by:  eqfnov2  5550