Theorem fresfo 46217

Description: Conditions for a restriction to be an onto function. Part of fresf1o 32288. (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 6578	. . . 4 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
2	1	biimpi 215	. . 3 ⊢ (Fun 𝐹 → 𝐹 Fn dom 𝐹)
3	2	adantr 480	. 2 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹) → 𝐹 Fn dom 𝐹)
4		sseqin2 4215	. . . . 5 ⊢ (𝐶 ⊆ ran 𝐹 ↔ (ran 𝐹 ∩ 𝐶) = 𝐶)
5	4	biimpi 215	. . . 4 ⊢ (𝐶 ⊆ ran 𝐹 → (ran 𝐹 ∩ 𝐶) = 𝐶)
6	5	eqcomd 2737	. . 3 ⊢ (𝐶 ⊆ ran 𝐹 → 𝐶 = (ran 𝐹 ∩ 𝐶))
7	6	adantl 481	. 2 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹) → 𝐶 = (ran 𝐹 ∩ 𝐶))
8		eqidd 2732	. 2 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹) → (◡𝐹 “ 𝐶) = (◡𝐹 “ 𝐶))
9	3, 7, 8	rescnvimafod 7075	1 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹) → (𝐹 ↾ (◡𝐹 “ 𝐶)):(◡𝐹 “ 𝐶)–onto→𝐶)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∩ cin 3947   ⊆ wss 3948  ^◡ccnv 5675  dom cdm 5676  ran crn 5677   ↾ cres 5678   “ cima 5679  Fun wfun 6537   Fn wfn 6538  –onto→wfo 6541

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-fun 6545  df-fn 6546  df-fo 6549

This theorem is referenced by: (None)