Theorem eqfnfv3 6943

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 6942	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))))
2		eqss 3941	. . . . 5 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
3	2	biancomi 464	. . . 4 ⊢ (𝐴 = 𝐵 ↔ (𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵))
4	3	anbi1i 625	. . 3 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)) ↔ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)))
5		anass 470	. . 3 ⊢ (((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)) ↔ (𝐵 ⊆ 𝐴 ∧ (𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))))
6		dfss3 3914	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)
7	6	anbi1i 625	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)) ↔ (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)))
8		r19.26 3111	. . . . 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 624	. . 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 205   ∧ wa 397   = wceq 1539   ∈ wcel 2104  ∀wral 3062   ⊆ wss 3892   Fn wfn 6453  ‘cfv 6458

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-iota 6410  df-fun 6460  df-fn 6461  df-fv 6466

This theorem is referenced by: (None)