Theorem rescnvimafod 7050

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 6068	. . . . 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 6622	. . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴)
6	5	eqcomd 2767	. . . 4 ⊢ (𝜑 → 𝐴 = dom 𝐹)
7	3, 4, 6	3sstr4d 3991	. . 3 ⊢ (𝜑 → 𝐷 ⊆ 𝐴)
8	1, 7	fnssresd 6641	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐷) Fn 𝐷)
9		df-ima 5658	. . . 4 ⊢ (𝐹 “ 𝐷) = ran (𝐹 ↾ 𝐷)
10	4	imaeq2d 6046	. . . . 5 ⊢ (𝜑 → (𝐹 “ 𝐷) = (𝐹 “ (◡𝐹 “ 𝐵)))
11		fnfun 6617	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
12		funimacnv 6598	. . . . . 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 4161	. . . . . 6 ⊢ (𝐵 ∩ ran 𝐹) = (ran 𝐹 ∩ 𝐵)
15	14	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐵 ∩ ran 𝐹) = (ran 𝐹 ∩ 𝐵))
16	10, 13, 15	3eqtrd 2800	. . . 4 ⊢ (𝜑 → (𝐹 “ 𝐷) = (ran 𝐹 ∩ 𝐵))
17	9, 16	eqtr3id 2810	. . 3 ⊢ (𝜑 → ran (𝐹 ↾ 𝐷) = (ran 𝐹 ∩ 𝐵))
18		rescnvimafod.e	. . 3 ⊢ (𝜑 → 𝐸 = (ran 𝐹 ∩ 𝐵))
19	17, 18	eqtr4d 2799	. 2 ⊢ (𝜑 → ran (𝐹 ↾ 𝐷) = 𝐸)
20		df-fo 6523	. 2 ⊢ ((𝐹 ↾ 𝐷):𝐷–onto→𝐸 ↔ ((𝐹 ↾ 𝐷) Fn 𝐷 ∧ ran (𝐹 ↾ 𝐷) = 𝐸))
21	8, 19, 20	sylanbrc 592	1 ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–onto→𝐸)

Syntax hints:   → wi 4   = wceq 1559   ∩ cin 3903   ⊆ wss 3904  ^◡ccnv 5644  dom cdm 5645  ran crn 5646   ↾ cres 5647   “ cima 5648  Fun wfun 6511   Fn wfn 6512  –onto→wfo 6515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-fun 6519  df-fn 6520  df-fo 6523

This theorem is referenced by:  fresfo  47606  fcoreslem3  47623  3f1oss1  47633