Theorem fresfo 47640

Description: Conditions for a restriction to be an onto function. Part of fresf1o 32888. (Contributed by AV, 29-Sep-2024.)

Assertion

Ref	Expression
fresfo	⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹) → (𝐹 ↾ (◡𝐹 “ 𝐶)):(◡𝐹 “ 𝐶)–onto→𝐶)

Proof of Theorem fresfo

Step	Hyp	Ref	Expression
1		funfn 6555	. . 3 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
2	1	birani 508	. 2 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹) → 𝐹 Fn dom 𝐹)
3		sseqin2 4178	. . . . 5 ⊢ (𝐶 ⊆ ran 𝐹 ↔ (ran 𝐹 ∩ 𝐶) = 𝐶)
4	3	biimpi 219	. . . 4 ⊢ (𝐶 ⊆ ran 𝐹 → (ran 𝐹 ∩ 𝐶) = 𝐶)
5	4	eqcomd 2771	. . 3 ⊢ (𝐶 ⊆ ran 𝐹 → 𝐶 = (ran 𝐹 ∩ 𝐶))
6	5	adantl 486	. 2 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹) → 𝐶 = (ran 𝐹 ∩ 𝐶))
7		eqidd 2766	. 2 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹) → (◡𝐹 “ 𝐶) = (◡𝐹 “ 𝐶))
8	2, 6, 7	rescnvimafod 7058	1 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹) → (𝐹 ↾ (◡𝐹 “ 𝐶)):(◡𝐹 “ 𝐶)–onto→𝐶)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∩ cin 3906   ⊆ wss 3907  ^◡ccnv 5651  dom cdm 5652  ran crn 5653   ↾ cres 5654   “ cima 5655  Fun wfun 6519   Fn wfn 6520  –onto→wfo 6523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-fun 6527  df-fn 6528  df-fo 6531

This theorem is referenced by: (None)