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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 6100 | . . . . 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 2743 | . . . 4 ⊢ (𝜑 → 𝐴 = dom 𝐹) |
| 7 | 3, 4, 6 | 3sstr4d 4039 | . . 3 ⊢ (𝜑 → 𝐷 ⊆ 𝐴) |
| 8 | 1, 7 | fnssresd 6692 | . 2 ⊢ (𝜑 → (𝐹 ↾ 𝐷) Fn 𝐷) |
| 9 | df-ima 5698 | . . . 4 ⊢ (𝐹 “ 𝐷) = ran (𝐹 ↾ 𝐷) | |
| 10 | 4 | imaeq2d 6078 | . . . . 5 ⊢ (𝜑 → (𝐹 “ 𝐷) = (𝐹 “ (◡𝐹 “ 𝐵))) |
| 11 | fnfun 6668 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹) | |
| 12 | funimacnv 6647 | . . . . . 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 4209 | . . . . . 6 ⊢ (𝐵 ∩ ran 𝐹) = (ran 𝐹 ∩ 𝐵) | |
| 15 | 14 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝐵 ∩ ran 𝐹) = (ran 𝐹 ∩ 𝐵)) |
| 16 | 10, 13, 15 | 3eqtrd 2781 | . . . 4 ⊢ (𝜑 → (𝐹 “ 𝐷) = (ran 𝐹 ∩ 𝐵)) |
| 17 | 9, 16 | eqtr3id 2791 | . . 3 ⊢ (𝜑 → ran (𝐹 ↾ 𝐷) = (ran 𝐹 ∩ 𝐵)) |
| 18 | rescnvimafod.e | . . 3 ⊢ (𝜑 → 𝐸 = (ran 𝐹 ∩ 𝐵)) | |
| 19 | 17, 18 | eqtr4d 2780 | . 2 ⊢ (𝜑 → ran (𝐹 ↾ 𝐷) = 𝐸) |
| 20 | df-fo 6567 | . 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) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∩ cin 3950 ⊆ wss 3951 ◡ccnv 5684 dom cdm 5685 ran crn 5686 ↾ cres 5687 “ cima 5688 Fun wfun 6555 Fn wfn 6556 –onto→wfo 6559 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-fun 6563 df-fn 6564 df-fo 6567 |
| This theorem is referenced by: fresfo 47060 fcoreslem3 47077 3f1oss1 47087 |
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