Theorem fresfo 44157

Description: Conditions for a restriction to be an onto function. Part of fresf1o 30639. (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 6388	. . . 4 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
2	1	biimpi 219	. . 3 ⊢ (Fun 𝐹 → 𝐹 Fn dom 𝐹)
3	2	adantr 484	. 2 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹) → 𝐹 Fn dom 𝐹)
4		sseqin2 4116	. . . . 5 ⊢ (𝐶 ⊆ ran 𝐹 ↔ (ran 𝐹 ∩ 𝐶) = 𝐶)
5	4	biimpi 219	. . . 4 ⊢ (𝐶 ⊆ ran 𝐹 → (ran 𝐹 ∩ 𝐶) = 𝐶)
6	5	eqcomd 2742	. . 3 ⊢ (𝐶 ⊆ ran 𝐹 → 𝐶 = (ran 𝐹 ∩ 𝐶))
7	6	adantl 485	. 2 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹) → 𝐶 = (ran 𝐹 ∩ 𝐶))
8		eqidd 2737	. 2 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹) → (◡𝐹 “ 𝐶) = (◡𝐹 “ 𝐶))
9	3, 7, 8	rescnvimafod 6872	1 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹) → (𝐹 ↾ (◡𝐹 “ 𝐶)):(◡𝐹 “ 𝐶)–onto→𝐶)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1543   ∩ cin 3852   ⊆ wss 3853  ^◡ccnv 5535  dom cdm 5536  ran crn 5537   ↾ cres 5538   “ cima 5539  Fun wfun 6352   Fn wfn 6353  –onto→wfo 6356

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-br 5040  df-opab 5102  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-fun 6360  df-fn 6361  df-fo 6364

This theorem is referenced by: (None)