Theorem rescnvimafod 7092

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 6101	. . . . 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 6673	. . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴)
6	5	eqcomd 2740	. . . 4 ⊢ (𝜑 → 𝐴 = dom 𝐹)
7	3, 4, 6	3sstr4d 4042	. . 3 ⊢ (𝜑 → 𝐷 ⊆ 𝐴)
8	1, 7	fnssresd 6692	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐷) Fn 𝐷)
9		df-ima 5701	. . . 4 ⊢ (𝐹 “ 𝐷) = ran (𝐹 ↾ 𝐷)
10	4	imaeq2d 6079	. . . . 5 ⊢ (𝜑 → (𝐹 “ 𝐷) = (𝐹 “ (◡𝐹 “ 𝐵)))
11		fnfun 6668	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
12		funimacnv 6648	. . . . . 6 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ 𝐵)) = (𝐵 ∩ ran 𝐹))
13	1, 11, 12	3syl 18	. . . . 5 ⊢ (𝜑 → (𝐹 “ (◡𝐹 “ 𝐵)) = (𝐵 ∩ ran 𝐹))
14		incom 4216	. . . . . 6 ⊢ (𝐵 ∩ ran 𝐹) = (ran 𝐹 ∩ 𝐵)
15	14	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐵 ∩ ran 𝐹) = (ran 𝐹 ∩ 𝐵))
16	10, 13, 15	3eqtrd 2778	. . . 4 ⊢ (𝜑 → (𝐹 “ 𝐷) = (ran 𝐹 ∩ 𝐵))
17	9, 16	eqtr3id 2788	. . 3 ⊢ (𝜑 → ran (𝐹 ↾ 𝐷) = (ran 𝐹 ∩ 𝐵))
18		rescnvimafod.e	. . 3 ⊢ (𝜑 → 𝐸 = (ran 𝐹 ∩ 𝐵))
19	17, 18	eqtr4d 2777	. 2 ⊢ (𝜑 → ran (𝐹 ↾ 𝐷) = 𝐸)
20		df-fo 6568	. 2 ⊢ ((𝐹 ↾ 𝐷):𝐷–onto→𝐸 ↔ ((𝐹 ↾ 𝐷) Fn 𝐷 ∧ ran (𝐹 ↾ 𝐷) = 𝐸))
21	8, 19, 20	sylanbrc 583	1 ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–onto→𝐸)

Syntax hints:   → wi 4   = wceq 1536   ∩ cin 3961   ⊆ wss 3962  ^◡ccnv 5687  dom cdm 5688  ran crn 5689   ↾ cres 5690   “ cima 5691  Fun wfun 6556   Fn wfn 6557  –onto→wfo 6560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-fun 6564  df-fn 6565  df-fo 6568

This theorem is referenced by:  fresfo  46997  fcoreslem3  47014  3f1oss1  47024