Theorem funsseq 35749

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 4054	. 2 ⊢ (𝐹 = 𝐺 → 𝐹 ⊆ 𝐺)
2		simpl3 1192	. . . . 5 ⊢ (((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) ∧ 𝐹 ⊆ 𝐺) → dom 𝐹 = dom 𝐺)
3	2	reseq2d 6000	. . . 4 ⊢ (((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) ∧ 𝐹 ⊆ 𝐺) → (𝐺 ↾ dom 𝐹) = (𝐺 ↾ dom 𝐺))
4		funssres 6612	. . . . 5 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺) → (𝐺 ↾ dom 𝐹) = 𝐹)
5	4	3ad2antl2 1185	. . . 4 ⊢ (((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) ∧ 𝐹 ⊆ 𝐺) → (𝐺 ↾ dom 𝐹) = 𝐹)
6		simpl2 1191	. . . . 5 ⊢ (((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) ∧ 𝐹 ⊆ 𝐺) → Fun 𝐺)
7		funrel 6585	. . . . 5 ⊢ (Fun 𝐺 → Rel 𝐺)
8		resdm 6046	. . . . 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 2783	. . 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 1537   ⊆ wss 3963  dom cdm 5689   ↾ cres 5691  Rel wrel 5694  Fun wfun 6557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-res 5701  df-fun 6565

This theorem is referenced by: (None)