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Mirrors > Home > MPE Home > Th. List > funimacnv | Structured version Visualization version GIF version |
Description: The image of the preimage of a function. (Contributed by NM, 25-May-2004.) |
Ref | Expression |
---|---|
funimacnv | ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ 𝐴)) = (𝐴 ∩ ran 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funcnvres2 6431 | . . . 4 ⊢ (Fun 𝐹 → ◡(◡𝐹 ↾ 𝐴) = (𝐹 ↾ (◡𝐹 “ 𝐴))) | |
2 | 1 | rneqd 5805 | . . 3 ⊢ (Fun 𝐹 → ran ◡(◡𝐹 ↾ 𝐴) = ran (𝐹 ↾ (◡𝐹 “ 𝐴))) |
3 | df-ima 5565 | . . 3 ⊢ (𝐹 “ (◡𝐹 “ 𝐴)) = ran (𝐹 ↾ (◡𝐹 “ 𝐴)) | |
4 | 2, 3 | syl6reqr 2874 | . 2 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ 𝐴)) = ran ◡(◡𝐹 ↾ 𝐴)) |
5 | df-rn 5563 | . . . 4 ⊢ ran 𝐹 = dom ◡𝐹 | |
6 | 5 | ineq2i 4183 | . . 3 ⊢ (𝐴 ∩ ran 𝐹) = (𝐴 ∩ dom ◡𝐹) |
7 | dmres 5872 | . . 3 ⊢ dom (◡𝐹 ↾ 𝐴) = (𝐴 ∩ dom ◡𝐹) | |
8 | dfdm4 5761 | . . 3 ⊢ dom (◡𝐹 ↾ 𝐴) = ran ◡(◡𝐹 ↾ 𝐴) | |
9 | 6, 7, 8 | 3eqtr2ri 2850 | . 2 ⊢ ran ◡(◡𝐹 ↾ 𝐴) = (𝐴 ∩ ran 𝐹) |
10 | 4, 9 | syl6eq 2871 | 1 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ 𝐴)) = (𝐴 ∩ ran 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∩ cin 3932 ◡ccnv 5551 dom cdm 5552 ran crn 5553 ↾ cres 5554 “ cima 5555 Fun wfun 6346 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5200 ax-nul 5207 ax-pr 5327 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3495 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4465 df-sn 4565 df-pr 4567 df-op 4571 df-br 5064 df-opab 5126 df-id 5457 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-rn 5563 df-res 5564 df-ima 5565 df-fun 6354 |
This theorem is referenced by: funimass1 6433 funimass2 6434 isercolllem2 15018 isercolllem3 15019 isercoll 15020 cncls 21878 preimane 30415 fnpreimac 30416 ffsrn 30465 cvmliftlem15 32569 imaelsetpreimafv 43629 |
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