Theorem fresfo 46965

Description: Conditions for a restriction to be an onto function. Part of fresf1o 32652. (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 6610	. . . 4 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
2	1	biimpi 216	. . 3 ⊢ (Fun 𝐹 → 𝐹 Fn dom 𝐹)
3	2	adantr 480	. 2 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹) → 𝐹 Fn dom 𝐹)
4		sseqin2 4244	. . . . 5 ⊢ (𝐶 ⊆ ran 𝐹 ↔ (ran 𝐹 ∩ 𝐶) = 𝐶)
5	4	biimpi 216	. . . 4 ⊢ (𝐶 ⊆ ran 𝐹 → (ran 𝐹 ∩ 𝐶) = 𝐶)
6	5	eqcomd 2746	. . 3 ⊢ (𝐶 ⊆ ran 𝐹 → 𝐶 = (ran 𝐹 ∩ 𝐶))
7	6	adantl 481	. 2 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹) → 𝐶 = (ran 𝐹 ∩ 𝐶))
8		eqidd 2741	. 2 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹) → (◡𝐹 “ 𝐶) = (◡𝐹 “ 𝐶))
9	3, 7, 8	rescnvimafod 7109	1 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹) → (𝐹 ↾ (◡𝐹 “ 𝐶)):(◡𝐹 “ 𝐶)–onto→𝐶)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∩ cin 3975   ⊆ wss 3976  ^◡ccnv 5699  dom cdm 5700  ran crn 5701   ↾ cres 5702   “ cima 5703  Fun wfun 6569   Fn wfn 6570  –onto→wfo 6573

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-rn 5711  df-res 5712  df-ima 5713  df-fun 6577  df-fn 6578  df-fo 6581

This theorem is referenced by: (None)