Theorem rescnvimafod 7077

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 6080	. . . . 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 6653	. . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴)
6	5	eqcomd 2731	. . . 4 ⊢ (𝜑 → 𝐴 = dom 𝐹)
7	3, 4, 6	3sstr4d 4020	. . 3 ⊢ (𝜑 → 𝐷 ⊆ 𝐴)
8	1, 7	fnssresd 6673	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐷) Fn 𝐷)
9		df-ima 5685	. . . 4 ⊢ (𝐹 “ 𝐷) = ran (𝐹 ↾ 𝐷)
10	4	imaeq2d 6058	. . . . 5 ⊢ (𝜑 → (𝐹 “ 𝐷) = (𝐹 “ (◡𝐹 “ 𝐵)))
11		fnfun 6648	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
12		funimacnv 6628	. . . . . 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 4195	. . . . . 6 ⊢ (𝐵 ∩ ran 𝐹) = (ran 𝐹 ∩ 𝐵)
15	14	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐵 ∩ ran 𝐹) = (ran 𝐹 ∩ 𝐵))
16	10, 13, 15	3eqtrd 2769	. . . 4 ⊢ (𝜑 → (𝐹 “ 𝐷) = (ran 𝐹 ∩ 𝐵))
17	9, 16	eqtr3id 2779	. . 3 ⊢ (𝜑 → ran (𝐹 ↾ 𝐷) = (ran 𝐹 ∩ 𝐵))
18		rescnvimafod.e	. . 3 ⊢ (𝜑 → 𝐸 = (ran 𝐹 ∩ 𝐵))
19	17, 18	eqtr4d 2768	. 2 ⊢ (𝜑 → ran (𝐹 ↾ 𝐷) = 𝐸)
20		df-fo 6548	. 2 ⊢ ((𝐹 ↾ 𝐷):𝐷–onto→𝐸 ↔ ((𝐹 ↾ 𝐷) Fn 𝐷 ∧ ran (𝐹 ↾ 𝐷) = 𝐸))
21	8, 19, 20	sylanbrc 581	1 ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–onto→𝐸)

Syntax hints:   → wi 4   = wceq 1533   ∩ cin 3939   ⊆ wss 3940  ^◡ccnv 5671  dom cdm 5672  ran crn 5673   ↾ cres 5674   “ cima 5675  Fun wfun 6536   Fn wfn 6537  –onto→wfo 6540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-fun 6544  df-fn 6545  df-fo 6548

This theorem is referenced by:  fresfo  46492  fcoreslem3  46509