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| Mirrors > Home > MPE Home > Th. List > cnvimainrn | Structured version Visualization version GIF version | ||
| Description: The preimage of the intersection of the range of a class and a class 𝐴 is the preimage of the class 𝐴. (Contributed by AV, 17-Sep-2024.) |
| Ref | Expression |
|---|---|
| cnvimainrn | ⊢ (Fun 𝐹 → (◡𝐹 “ (ran 𝐹 ∩ 𝐴)) = (◡𝐹 “ 𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | inpreima 7084 | . 2 ⊢ (Fun 𝐹 → (◡𝐹 “ (ran 𝐹 ∩ 𝐴)) = ((◡𝐹 “ ran 𝐹) ∩ (◡𝐹 “ 𝐴))) | |
| 2 | cnvimass 6100 | . . . . 5 ⊢ (◡𝐹 “ 𝐴) ⊆ dom 𝐹 | |
| 3 | cnvimarndm 6101 | . . . . 5 ⊢ (◡𝐹 “ ran 𝐹) = dom 𝐹 | |
| 4 | 2, 3 | sseqtrri 4033 | . . . 4 ⊢ (◡𝐹 “ 𝐴) ⊆ (◡𝐹 “ ran 𝐹) |
| 5 | dfss2 3969 | . . . 4 ⊢ ((◡𝐹 “ 𝐴) ⊆ (◡𝐹 “ ran 𝐹) ↔ ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ ran 𝐹)) = (◡𝐹 “ 𝐴)) | |
| 6 | 4, 5 | mpbi 230 | . . 3 ⊢ ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ ran 𝐹)) = (◡𝐹 “ 𝐴) |
| 7 | 6 | ineqcomi 4211 | . 2 ⊢ ((◡𝐹 “ ran 𝐹) ∩ (◡𝐹 “ 𝐴)) = (◡𝐹 “ 𝐴) |
| 8 | 1, 7 | eqtrdi 2793 | 1 ⊢ (Fun 𝐹 → (◡𝐹 “ (ran 𝐹 ∩ 𝐴)) = (◡𝐹 “ 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∩ cin 3950 ⊆ wss 3951 ◡ccnv 5684 dom cdm 5685 ran crn 5686 “ cima 5688 Fun wfun 6555 |
| 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-10 2141 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-nf 1784 df-sb 2065 df-mo 2540 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 |
| This theorem is referenced by: fcoreslem1 47075 |
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