Theorem eqfnov 7484

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 6983	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐺 Fn (𝐶 × 𝐷)) → (𝐹 = 𝐺 ↔ ((𝐴 × 𝐵) = (𝐶 × 𝐷) ∧ ∀𝑧 ∈ (𝐴 × 𝐵)(𝐹‘𝑧) = (𝐺‘𝑧))))
2		fveq2 6842	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
3		fveq2 6842	. . . . . 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 7359	. . . . . 6 ⊢ (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
6		df-ov 7359	. . . . . 6 ⊢ (𝑥𝐺𝑦) = (𝐺‘⟨𝑥, 𝑦⟩)
7	5, 6	eqeq12i 2754	. . . . 5 ⊢ ((𝑥𝐹𝑦) = (𝑥𝐺𝑦) ↔ (𝐹‘⟨𝑥, 𝑦⟩) = (𝐺‘⟨𝑥, 𝑦⟩))
8	4, 7	bitr4di 288	. . . 4 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐹‘𝑧) = (𝐺‘𝑧) ↔ (𝑥𝐹𝑦) = (𝑥𝐺𝑦)))
9	8	ralxp 5797	. . 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 286	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 205   ∧ wa 396   = wceq 1541  ∀wral 3064  ⟨cop 4592   × cxp 5631   Fn wfn 6491  ‘cfv 6496  (class class class)co 7356

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-fv 6504  df-ov 7359

This theorem is referenced by:  eqfnov2  7485  oprres  7521  ssceq  17708  sspg  29617  ssps  29619  sspmlem  29621