Theorem funsseq 35762

Description: Given two functions with equal domains, equality only requires one direction of the subset relationship. (Contributed by Scott Fenton, 24-Apr-2012.) (Proof shortened by Mario Carneiro, 3-May-2015.)

Assertion

Ref	Expression
funsseq	⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) → (𝐹 = 𝐺 ↔ 𝐹 ⊆ 𝐺))

Proof of Theorem funsseq

Step	Hyp	Ref	Expression
1		eqimss 4008	. 2 ⊢ (𝐹 = 𝐺 → 𝐹 ⊆ 𝐺)
2		simpl3 1194	. . . . 5 ⊢ (((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) ∧ 𝐹 ⊆ 𝐺) → dom 𝐹 = dom 𝐺)
3	2	reseq2d 5953	. . . 4 ⊢ (((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) ∧ 𝐹 ⊆ 𝐺) → (𝐺 ↾ dom 𝐹) = (𝐺 ↾ dom 𝐺))
4		funssres 6563	. . . . 5 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺) → (𝐺 ↾ dom 𝐹) = 𝐹)
5	4	3ad2antl2 1187	. . . 4 ⊢ (((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) ∧ 𝐹 ⊆ 𝐺) → (𝐺 ↾ dom 𝐹) = 𝐹)
6		simpl2 1193	. . . . 5 ⊢ (((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) ∧ 𝐹 ⊆ 𝐺) → Fun 𝐺)
7		funrel 6536	. . . . 5 ⊢ (Fun 𝐺 → Rel 𝐺)
8		resdm 6000	. . . . 5 ⊢ (Rel 𝐺 → (𝐺 ↾ dom 𝐺) = 𝐺)
9	6, 7, 8	3syl 18	. . . 4 ⊢ (((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) ∧ 𝐹 ⊆ 𝐺) → (𝐺 ↾ dom 𝐺) = 𝐺)
10	3, 5, 9	3eqtr3d 2773	. . 3 ⊢ (((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) ∧ 𝐹 ⊆ 𝐺) → 𝐹 = 𝐺)
11	10	ex 412	. 2 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) → (𝐹 ⊆ 𝐺 → 𝐹 = 𝐺))
12	1, 11	impbid2 226	1 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) → (𝐹 = 𝐺 ↔ 𝐹 ⊆ 𝐺))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ⊆ wss 3917  dom cdm 5641   ↾ cres 5643  Rel wrel 5646  Fun wfun 6508

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-res 5653  df-fun 6516

This theorem is referenced by: (None)