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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 7083 | . 2 ⊢ (Fun 𝐹 → (◡𝐹 “ (ran 𝐹 ∩ 𝐴)) = ((◡𝐹 “ ran 𝐹) ∩ (◡𝐹 “ 𝐴))) | |
2 | cnvimass 6101 | . . . . 5 ⊢ (◡𝐹 “ 𝐴) ⊆ dom 𝐹 | |
3 | cnvimarndm 6102 | . . . . 5 ⊢ (◡𝐹 “ ran 𝐹) = dom 𝐹 | |
4 | 2, 3 | sseqtrri 4032 | . . . 4 ⊢ (◡𝐹 “ 𝐴) ⊆ (◡𝐹 “ ran 𝐹) |
5 | dfss2 3980 | . . . 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 4218 | . 2 ⊢ ((◡𝐹 “ ran 𝐹) ∩ (◡𝐹 “ 𝐴)) = (◡𝐹 “ 𝐴) |
8 | 1, 7 | eqtrdi 2790 | 1 ⊢ (Fun 𝐹 → (◡𝐹 “ (ran 𝐹 ∩ 𝐴)) = (◡𝐹 “ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∩ cin 3961 ⊆ wss 3962 ◡ccnv 5687 dom cdm 5688 ran crn 5689 “ cima 5691 Fun wfun 6556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-fun 6564 |
This theorem is referenced by: fcoreslem1 47012 |
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