Theorem eqfnov 7487

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 6977	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐺 Fn (𝐶 × 𝐷)) → (𝐹 = 𝐺 ↔ ((𝐴 × 𝐵) = (𝐶 × 𝐷) ∧ ∀𝑧 ∈ (𝐴 × 𝐵)(𝐹‘𝑧) = (𝐺‘𝑧))))
2		fveq2 6834	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
3		fveq2 6834	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺‘𝑧) = (𝐺‘⟨𝑥, 𝑦⟩))
4	2, 3	eqeq12d 2752	. . . . 5 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐹‘𝑧) = (𝐺‘𝑧) ↔ (𝐹‘⟨𝑥, 𝑦⟩) = (𝐺‘⟨𝑥, 𝑦⟩)))
5		df-ov 7361	. . . . . 6 ⊢ (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
6		df-ov 7361	. . . . . 6 ⊢ (𝑥𝐺𝑦) = (𝐺‘⟨𝑥, 𝑦⟩)
7	5, 6	eqeq12i 2754	. . . . 5 ⊢ ((𝑥𝐹𝑦) = (𝑥𝐺𝑦) ↔ (𝐹‘⟨𝑥, 𝑦⟩) = (𝐺‘⟨𝑥, 𝑦⟩))
8	4, 7	bitr4di 289	. . . 4 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐹‘𝑧) = (𝐺‘𝑧) ↔ (𝑥𝐹𝑦) = (𝑥𝐺𝑦)))
9	8	ralxp 5790	. . 3 ⊢ (∀𝑧 ∈ (𝐴 × 𝐵)(𝐹‘𝑧) = (𝐺‘𝑧) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦))
10	9	anbi2i 623	. 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	bitrdi 287	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   ↔ wb 206   ∧ wa 395   = wceq 1541  ∀wral 3051  ⟨cop 4586   × cxp 5622   Fn wfn 6487  ‘cfv 6492  (class class class)co 7358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-fv 6500  df-ov 7361

This theorem is referenced by:  eqfnov2  7488  oprres  7526  ssceq  17750  sspg  30803  ssps  30805  sspmlem  30807