Theorem eqfnfv3 6975

Description: Derive equality of functions from equality of their values. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
eqfnfv3	⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ (𝐹‘𝑥) = (𝐺‘𝑥)))))

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

Proof of Theorem eqfnfv3

Step	Hyp	Ref	Expression
1		eqfnfv2 6974	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))))
2		eqss 3946	. . . . 5 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
3	2	biancomi 462	. . . 4 ⊢ (𝐴 = 𝐵 ↔ (𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵))
4	3	anbi1i 624	. . 3 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)) ↔ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)))
5		anass 468	. . 3 ⊢ (((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)) ↔ (𝐵 ⊆ 𝐴 ∧ (𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))))
6		dfss3 3919	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)
7	6	anbi1i 624	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)) ↔ (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)))
8		r19.26 3093	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ (𝐹‘𝑥) = (𝐺‘𝑥)) ↔ (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)))
9	7, 8	bitr4i 278	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ (𝐹‘𝑥) = (𝐺‘𝑥)))
10	9	anbi2i 623	. . 3 ⊢ ((𝐵 ⊆ 𝐴 ∧ (𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))) ↔ (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ (𝐹‘𝑥) = (𝐺‘𝑥))))
11	4, 5, 10	3bitri 297	. 2 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)) ↔ (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ (𝐹‘𝑥) = (𝐺‘𝑥))))
12	1, 11	bitrdi 287	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ (𝐹‘𝑥) = (𝐺‘𝑥)))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3048   ⊆ wss 3898   Fn wfn 6484  ‘cfv 6489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  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

This theorem is referenced by: (None)