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Theorem eqfnov 7562
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) ∧ ∀𝑥𝐴𝑦𝐵 (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.
StepHypRef Expression
1 eqfnfv2 7052 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐺 Fn (𝐶 × 𝐷)) → (𝐹 = 𝐺 ↔ ((𝐴 × 𝐵) = (𝐶 × 𝐷) ∧ ∀𝑧 ∈ (𝐴 × 𝐵)(𝐹𝑧) = (𝐺𝑧))))
2 fveq2 6906 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
3 fveq2 6906 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺𝑧) = (𝐺‘⟨𝑥, 𝑦⟩))
42, 3eqeq12d 2753 . . . . 5 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐹𝑧) = (𝐺𝑧) ↔ (𝐹‘⟨𝑥, 𝑦⟩) = (𝐺‘⟨𝑥, 𝑦⟩)))
5 df-ov 7434 . . . . . 6 (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
6 df-ov 7434 . . . . . 6 (𝑥𝐺𝑦) = (𝐺‘⟨𝑥, 𝑦⟩)
75, 6eqeq12i 2755 . . . . 5 ((𝑥𝐹𝑦) = (𝑥𝐺𝑦) ↔ (𝐹‘⟨𝑥, 𝑦⟩) = (𝐺‘⟨𝑥, 𝑦⟩))
84, 7bitr4di 289 . . . 4 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐹𝑧) = (𝐺𝑧) ↔ (𝑥𝐹𝑦) = (𝑥𝐺𝑦)))
98ralxp 5852 . . 3 (∀𝑧 ∈ (𝐴 × 𝐵)(𝐹𝑧) = (𝐺𝑧) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦))
109anbi2i 623 . 2 (((𝐴 × 𝐵) = (𝐶 × 𝐷) ∧ ∀𝑧 ∈ (𝐴 × 𝐵)(𝐹𝑧) = (𝐺𝑧)) ↔ ((𝐴 × 𝐵) = (𝐶 × 𝐷) ∧ ∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦)))
111, 10bitrdi 287 1 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐺 Fn (𝐶 × 𝐷)) → (𝐹 = 𝐺 ↔ ((𝐴 × 𝐵) = (𝐶 × 𝐷) ∧ ∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wral 3061  cop 4632   × cxp 5683   Fn wfn 6556  cfv 6561  (class class class)co 7431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569  df-ov 7434
This theorem is referenced by:  eqfnov2  7563  oprres  7601  ssceq  17870  sspg  30747  ssps  30749  sspmlem  30751
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