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Theorem eqfnov 6719
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 6268 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐺 Fn (𝐶 × 𝐷)) → (𝐹 = 𝐺 ↔ ((𝐴 × 𝐵) = (𝐶 × 𝐷) ∧ ∀𝑧 ∈ (𝐴 × 𝐵)(𝐹𝑧) = (𝐺𝑧))))
2 fveq2 6148 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
3 fveq2 6148 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺𝑧) = (𝐺‘⟨𝑥, 𝑦⟩))
42, 3eqeq12d 2636 . . . . 5 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐹𝑧) = (𝐺𝑧) ↔ (𝐹‘⟨𝑥, 𝑦⟩) = (𝐺‘⟨𝑥, 𝑦⟩)))
5 df-ov 6607 . . . . . 6 (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
6 df-ov 6607 . . . . . 6 (𝑥𝐺𝑦) = (𝐺‘⟨𝑥, 𝑦⟩)
75, 6eqeq12i 2635 . . . . 5 ((𝑥𝐹𝑦) = (𝑥𝐺𝑦) ↔ (𝐹‘⟨𝑥, 𝑦⟩) = (𝐺‘⟨𝑥, 𝑦⟩))
84, 7syl6bbr 278 . . . 4 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐹𝑧) = (𝐺𝑧) ↔ (𝑥𝐹𝑦) = (𝑥𝐺𝑦)))
98ralxp 5223 . . 3 (∀𝑧 ∈ (𝐴 × 𝐵)(𝐹𝑧) = (𝐺𝑧) ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦))
109anbi2i 729 . 2 (((𝐴 × 𝐵) = (𝐶 × 𝐷) ∧ ∀𝑧 ∈ (𝐴 × 𝐵)(𝐹𝑧) = (𝐺𝑧)) ↔ ((𝐴 × 𝐵) = (𝐶 × 𝐷) ∧ ∀𝑥𝐴𝑦𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦)))
111, 10syl6bb 276 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 196  wa 384   = wceq 1480  wral 2907  cop 4154   × cxp 5072   Fn wfn 5842  cfv 5847  (class class class)co 6604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-fv 5855  df-ov 6607
This theorem is referenced by:  eqfnov2  6720  oprres  6755  ssceq  16407  sspg  27432  ssps  27434  sspmlem  27436
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