Theorem eqfnov2 6665

Description: Two operators with the same domain are equal iff their values at each point in the domain are equal. (Contributed by Jeff Madsen, 7-Jun-2010.)

Assertion

Ref	Expression
eqfnov2	⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐺 Fn (𝐴 × 𝐵)) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦)))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦

Proof of Theorem eqfnov2

Step	Hyp	Ref	Expression
1		eqfnov 6664	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐺 Fn (𝐴 × 𝐵)) → (𝐹 = 𝐺 ↔ ((𝐴 × 𝐵) = (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦))))
2		simpr 476	. . 3 ⊢ (((𝐴 × 𝐵) = (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦))
3		eqidd 2611	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦) → (𝐴 × 𝐵) = (𝐴 × 𝐵))
4	3	ancri 573	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦) → ((𝐴 × 𝐵) = (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦)))
5	2, 4	impbii 198	. 2 ⊢ (((𝐴 × 𝐵) = (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦))
6	1, 5	syl6bb 275	1 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐺 Fn (𝐴 × 𝐵)) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∀wral 2896   × cxp 5036   Fn wfn 5799  (class class class)co 6549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-ov 6552

This theorem is referenced by:  fnmpt2ovd  7139  tpossym  7271  uncfcurf  16702  mamuass  20027  mamudi  20028  mamudir  20029  mamuvs1  20030  mamuvs2  20031  eqmat  20049  mamulid  20066  mamurid  20067  madutpos  20267  ressprdsds  21986  isngp3  22212  xrsdsre  22421  hhip  27418  1smat1  29198  matunitlindflem2  32576