Theorem funsseq 35800

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 3993	. 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 5928	. . . 4 ⊢ (((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) ∧ 𝐹 ⊆ 𝐺) → (𝐺 ↾ dom 𝐹) = (𝐺 ↾ dom 𝐺))
4		funssres 6525	. . . . 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 6498	. . . . 5 ⊢ (Fun 𝐺 → Rel 𝐺)
8		resdm 5975	. . . . 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 2774	. . 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 1541   ⊆ wss 3902  dom cdm 5616   ↾ cres 5618  Rel wrel 5621  Fun wfun 6475

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 2113  ax-9 2121  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

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-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-res 5628  df-fun 6483

This theorem is referenced by: (None)