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Theorem eqfnov 5733
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 5382 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐺 Fn (𝐶 × 𝐷)) → (𝐹 = 𝐺 ↔ ((𝐴 × 𝐵) = (𝐶 × 𝐷) ∧ ∀𝑧 ∈ (𝐴 × 𝐵)(𝐹𝑧) = (𝐺𝑧))))
2 fveq2 5289 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
3 fveq2 5289 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺𝑧) = (𝐺‘⟨𝑥, 𝑦⟩))
42, 3eqeq12d 2102 . . . . 5 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐹𝑧) = (𝐺𝑧) ↔ (𝐹‘⟨𝑥, 𝑦⟩) = (𝐺‘⟨𝑥, 𝑦⟩)))
5 df-ov 5637 . . . . . 6 (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
6 df-ov 5637 . . . . . 6 (𝑥𝐺𝑦) = (𝐺‘⟨𝑥, 𝑦⟩)
75, 6eqeq12i 2101 . . . . 5 ((𝑥𝐹𝑦) = (𝑥𝐺𝑦) ↔ (𝐹‘⟨𝑥, 𝑦⟩) = (𝐺‘⟨𝑥, 𝑦⟩))
84, 7syl6bbr 196 . . . 4 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐹𝑧) = (𝐺𝑧) ↔ (𝑥𝐹𝑦) = (𝑥𝐺𝑦)))
98ralxp 4567 . . 3 (∀𝑧 ∈ (𝐴 × 𝐵)(𝐹𝑧) = (𝐺𝑧) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦))
109anbi2i 445 . 2 (((𝐴 × 𝐵) = (𝐶 × 𝐷) ∧ ∀𝑧 ∈ (𝐴 × 𝐵)(𝐹𝑧) = (𝐺𝑧)) ↔ ((𝐴 × 𝐵) = (𝐶 × 𝐷) ∧ ∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦)))
111, 10syl6bb 194 1 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐺 Fn (𝐶 × 𝐷)) → (𝐹 = 𝐺 ↔ ((𝐴 × 𝐵) = (𝐶 × 𝐷) ∧ ∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  wral 2359  cop 3444   × cxp 4426   Fn wfn 4997  cfv 5002  (class class class)co 5634
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-csb 2932  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fn 5005  df-fv 5010  df-ov 5637
This theorem is referenced by:  eqfnov2  5734
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