Theorem fresfo 48062

Description: Conditions for a restriction to be an onto function. Part of fresf1o 33207. (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 6562	. . 3 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
2	1	birani 509	. 2 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹) → 𝐹 Fn dom 𝐹)
3		sseqin2 4169	. . . . 5 ⊢ (𝐶 ⊆ ran 𝐹 ↔ (ran 𝐹 ∩ 𝐶) = 𝐶)
4	3	biimpi 219	. . . 4 ⊢ (𝐶 ⊆ ran 𝐹 → (ran 𝐹 ∩ 𝐶) = 𝐶)
5	4	eqcomd 2767	. . 3 ⊢ (𝐶 ⊆ ran 𝐹 → 𝐶 = (ran 𝐹 ∩ 𝐶))
6	5	adantl 487	. 2 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹) → 𝐶 = (ran 𝐹 ∩ 𝐶))
7		eqidd 2762	. 2 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹) → (◡𝐹 “ 𝐶) = (◡𝐹 “ 𝐶))
8	2, 6, 7	rescnvimafod 7065	1 ⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹) → (𝐹 ↾ (◡𝐹 “ 𝐶)):(◡𝐹 “ 𝐶)–onto→𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∩ cin 3898   ⊆ wss 3899  ^◡ccnv 5650  dom cdm 5651  ran crn 5652   ↾ cres 5653   “ cima 5654  Fun wfun 6525   Fn wfn 6526  –onto→wfo 6529

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-fun 6533  df-fn 6534  df-fo 6537

This theorem is used by: (None)