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Mirrors > Home > MPE Home > Th. List > rescnvimafod | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
rescnvimafod.f | ⊢ (𝜑 → 𝐹 Fn 𝐴) |
rescnvimafod.e | ⊢ (𝜑 → 𝐸 = (ran 𝐹 ∩ 𝐵)) |
rescnvimafod.d | ⊢ (𝜑 → 𝐷 = (◡𝐹 “ 𝐵)) |
Ref | Expression |
---|---|
rescnvimafod | ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–onto→𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rescnvimafod.f | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) | |
2 | cnvimass 5934 | . . . . 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 6461 | . . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
6 | 5 | eqcomd 2742 | . . . 4 ⊢ (𝜑 → 𝐴 = dom 𝐹) |
7 | 3, 4, 6 | 3sstr4d 3934 | . . 3 ⊢ (𝜑 → 𝐷 ⊆ 𝐴) |
8 | 1, 7 | fnssresd 6479 | . 2 ⊢ (𝜑 → (𝐹 ↾ 𝐷) Fn 𝐷) |
9 | df-ima 5549 | . . . 4 ⊢ (𝐹 “ 𝐷) = ran (𝐹 ↾ 𝐷) | |
10 | 4 | imaeq2d 5914 | . . . . 5 ⊢ (𝜑 → (𝐹 “ 𝐷) = (𝐹 “ (◡𝐹 “ 𝐵))) |
11 | fnfun 6457 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹) | |
12 | funimacnv 6439 | . . . . . 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 4101 | . . . . . 6 ⊢ (𝐵 ∩ ran 𝐹) = (ran 𝐹 ∩ 𝐵) | |
15 | 14 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝐵 ∩ ran 𝐹) = (ran 𝐹 ∩ 𝐵)) |
16 | 10, 13, 15 | 3eqtrd 2775 | . . . 4 ⊢ (𝜑 → (𝐹 “ 𝐷) = (ran 𝐹 ∩ 𝐵)) |
17 | 9, 16 | eqtr3id 2785 | . . 3 ⊢ (𝜑 → ran (𝐹 ↾ 𝐷) = (ran 𝐹 ∩ 𝐵)) |
18 | rescnvimafod.e | . . 3 ⊢ (𝜑 → 𝐸 = (ran 𝐹 ∩ 𝐵)) | |
19 | 17, 18 | eqtr4d 2774 | . 2 ⊢ (𝜑 → ran (𝐹 ↾ 𝐷) = 𝐸) |
20 | df-fo 6364 | . 2 ⊢ ((𝐹 ↾ 𝐷):𝐷–onto→𝐸 ↔ ((𝐹 ↾ 𝐷) Fn 𝐷 ∧ ran (𝐹 ↾ 𝐷) = 𝐸)) | |
21 | 8, 19, 20 | sylanbrc 586 | 1 ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–onto→𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∩ cin 3852 ⊆ wss 3853 ◡ccnv 5535 dom cdm 5536 ran crn 5537 ↾ cres 5538 “ cima 5539 Fun wfun 6352 Fn wfn 6353 –onto→wfo 6356 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-fun 6360 df-fn 6361 df-fo 6364 |
This theorem is referenced by: fresfo 44157 fcoreslem3 44174 |
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