Theorem rescnvimafod 7074

Description: The restriction of a function to a preimage of a class is a function onto the intersection of this class and the range of the function. (Contributed by AV, 13-Sep-2024.) (Revised by AV, 29-Sep-2024.)

Hypotheses

Ref	Expression
rescnvimafod.f	⊢ (𝜑 → 𝐹 Fn 𝐴)
rescnvimafod.e	⊢ (𝜑 → 𝐸 = (ran 𝐹 ∩ 𝐵))
rescnvimafod.d	⊢ (𝜑 → 𝐷 = (◡𝐹 “ 𝐵))

Assertion

Ref	Expression
rescnvimafod	⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–onto→𝐸)

Proof of Theorem rescnvimafod

Step	Hyp	Ref	Expression
1		rescnvimafod.f	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
2		cnvimass 6198	. . . . 5 ⊢ (◡𝐹 “ 𝐵) ⊆ dom 𝐹
3	2	a1i 11	. . . 4 ⊢ (𝜑 → (◡𝐹 “ 𝐵) ⊆ dom 𝐹)
4		rescnvimafod.d	. . . 4 ⊢ (𝜑 → 𝐷 = (◡𝐹 “ 𝐵))
5	1	fndmd 6645	. . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴)
6	5	eqcomd 2766	. . . 4 ⊢ (𝜑 → 𝐴 = dom 𝐹)
7	3, 4, 6	3sstr4d 3963	. . 3 ⊢ (𝜑 → 𝐷 ⊆ 𝐴)
8	1, 7	fnssresd 6664	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐷) Fn 𝐷)
9		df-ima 5663	. . . 4 ⊢ (𝐹 “ 𝐷) = ran (𝐹 ↾ 𝐷)
10	4	imaeq2d 6051	. . . . 5 ⊢ (𝜑 → (𝐹 “ 𝐷) = (𝐹 “ (◡𝐹 “ 𝐵)))
11		fnfun 6640	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
12		funimacnv 6622	. . . . . 6 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ 𝐵)) = (𝐵 ∩ ran 𝐹))
13	1, 11, 12	3syl 19	. . . . 5 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ 𝐵)) = (𝐵 ∩ ran 𝐹))
14		incom 4106	. . . . . 6 ⊢ (𝐵 ∩ ran 𝐹) = (ran 𝐹 ∩ 𝐵)
15	14	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐵 ∩ ran 𝐹) = (ran 𝐹 ∩ 𝐵))
16	10, 13, 15	3eqtrd 2799	. . . 4 ⊢ (𝜑 → (𝐹 “ 𝐷) = (ran 𝐹 ∩ 𝐵))
17	9, 16	eqtr3id 2809	. . 3 ⊢ (𝜑 → ran (𝐹 ↾ 𝐷) = (ran 𝐹 ∩ 𝐵))
18		rescnvimafod.e	. . 3 ⊢ (𝜑 → 𝐸 = (ran 𝐹 ∩ 𝐵))
19	17, 18	eqtr4d 2798	. 2 ⊢ (𝜑 → ran (𝐹 ↾ 𝐷) = 𝐸)
20		df-fo 6545	. 2 ⊢ ((𝐹 ↾ 𝐷):𝐷–onto→𝐸 ↔ ((𝐹 ↾ 𝐷) Fn 𝐷 ∧ ran (𝐹 ↾ 𝐷) = 𝐸))
21	8, 19, 20	sylanbrc 595	1 ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–onto→𝐸)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ⊆ wss 3895   ∩ cin 4089  ^◡ccnv 5649  dom cdm 5650  ran crn 5651   ↾ cres 5652   “ cima 5653  Fun wfun 6533   Fn wfn 6534  –onto→wfo 6537

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-fun 6541  df-fn 6542  df-fo 6545

This theorem is used by:  fresfo  48212  fcoreslem3  48229  3f1oss1  48239