Theorem funsseq 32005

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 3807	. 2 ⊢ (𝐹 = 𝐺 → 𝐹 ⊆ 𝐺)
2		simpl3 1231	. . . . 5 ⊢ (((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) ∧ 𝐹 ⊆ 𝐺) → dom 𝐹 = dom 𝐺)
3	2	reseq2d 5535	. . . 4 ⊢ (((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) ∧ 𝐹 ⊆ 𝐺) → (𝐺 ↾ dom 𝐹) = (𝐺 ↾ dom 𝐺))
4		funssres 6074	. . . . 5 ⊢ ((Fun 𝐺 ∧ 𝐹 ⊆ 𝐺) → (𝐺 ↾ dom 𝐹) = 𝐹)
5	4	3ad2antl2 1201	. . . 4 ⊢ (((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) ∧ 𝐹 ⊆ 𝐺) → (𝐺 ↾ dom 𝐹) = 𝐹)
6		simpl2 1229	. . . . 5 ⊢ (((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) ∧ 𝐹 ⊆ 𝐺) → Fun 𝐺)
7		funrel 6049	. . . . 5 ⊢ (Fun 𝐺 → Rel 𝐺)
8		resdm 5583	. . . . 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 2813	. . 3 ⊢ (((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) ∧ 𝐹 ⊆ 𝐺) → 𝐹 = 𝐺)
11	10	ex 397	. 2 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) → (𝐹 ⊆ 𝐺 → 𝐹 = 𝐺))
12	1, 11	impbid2 216	1 ⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) → (𝐹 = 𝐺 ↔ 𝐹 ⊆ 𝐺))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   ∧ w3a 1071   = wceq 1631   ⊆ wss 3724  dom cdm 5250   ↾ cres 5252  Rel wrel 5255  Fun wfun 6026

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-res 5262  df-fun 6034

This theorem is referenced by: (None)