Theorem eqfnov2 7490

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 7489	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐺 Fn (𝐴 × 𝐵)) → (𝐹 = 𝐺 ↔ ((𝐴 × 𝐵) = (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦))))
2		simpr 486	. . 3 ⊢ (((𝐴 × 𝐵) = (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦))
3		eqidd 2742	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦) → (𝐴 × 𝐵) = (𝐴 × 𝐵))
4	3	ancri 555	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦) → ((𝐴 × 𝐵) = (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦)))
5	2, 4	impbii 211	. 2 ⊢ (((𝐴 × 𝐵) = (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦))
6	1, 5	bitrdi 289	1 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐺 Fn (𝐴 × 𝐵)) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   = wceq 1548  ∀wral 3055   × cxp 5619   Fn wfn 6484  (class class class)co 7360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-fv 6497  df-ov 7363

This theorem is referenced by:  fnmpoovd  8030  tpossym  8202  uncfcurf  18200  mamuass  22389  mamudi  22390  mamudir  22391  mamuvs1  22392  mamuvs2  22393  eqmat  22411  mamulid  22428  mamurid  22429  madutpos  22629  ressprdsds  24358  isngp3  24585  xrsdsre  24798  hhip  31270  1smat1  34000  matunitlindflem2  37999  eqfnovd  49370  functhinclem1  49948