Theorem funsseq 35731

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 4067	. 2 ⊢ (𝐹 = 𝐺 → 𝐹 ⊆ 𝐺)
2		simpl3 1193	. . . . 5 ⊢ (((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) ∧ 𝐹 ⊆ 𝐺) → dom 𝐹 = dom 𝐺)
3	2	reseq2d 6009	. . . 4 ⊢ (((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) ∧ 𝐹 ⊆ 𝐺) → (𝐺 ↾ dom 𝐹) = (𝐺 ↾ dom 𝐺))
4		funssres 6622	. . . . 5 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺) → (𝐺 ↾ dom 𝐹) = 𝐹)
5	4	3ad2antl2 1186	. . . 4 ⊢ (((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) ∧ 𝐹 ⊆ 𝐺) → (𝐺 ↾ dom 𝐹) = 𝐹)
6		simpl2 1192	. . . . 5 ⊢ (((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) ∧ 𝐹 ⊆ 𝐺) → Fun 𝐺)
7		funrel 6595	. . . . 5 ⊢ (Fun 𝐺 → Rel 𝐺)
8		resdm 6055	. . . . 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 2788	. . 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 1087   = wceq 1537   ⊆ wss 3976  dom cdm 5700   ↾ cres 5702  Rel wrel 5705  Fun wfun 6567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-res 5712  df-fun 6575

This theorem is referenced by: (None)