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Theorem cnvimassrndm 6152
Description: The preimage of a superset of the range of a class is the domain of the class. Generalization of cnvimarndm 6082 for subsets. (Contributed by AV, 18-Sep-2024.)
Assertion
Ref Expression
cnvimassrndm (ran 𝐹𝐴 → (𝐹𝐴) = dom 𝐹)

Proof of Theorem cnvimassrndm
StepHypRef Expression
1 ssequn1 4175 . 2 (ran 𝐹𝐴 ↔ (ran 𝐹𝐴) = 𝐴)
2 imaeq2 6055 . . . . 5 (𝐴 = (ran 𝐹𝐴) → (𝐹𝐴) = (𝐹 “ (ran 𝐹𝐴)))
3 imaundi 6150 . . . . 5 (𝐹 “ (ran 𝐹𝐴)) = ((𝐹 “ ran 𝐹) ∪ (𝐹𝐴))
42, 3eqtrdi 2781 . . . 4 (𝐴 = (ran 𝐹𝐴) → (𝐹𝐴) = ((𝐹 “ ran 𝐹) ∪ (𝐹𝐴)))
5 cnvimarndm 6082 . . . . . 6 (𝐹 “ ran 𝐹) = dom 𝐹
65uneq1i 4153 . . . . 5 ((𝐹 “ ran 𝐹) ∪ (𝐹𝐴)) = (dom 𝐹 ∪ (𝐹𝐴))
7 cnvimass 6081 . . . . . 6 (𝐹𝐴) ⊆ dom 𝐹
8 ssequn2 4178 . . . . . 6 ((𝐹𝐴) ⊆ dom 𝐹 ↔ (dom 𝐹 ∪ (𝐹𝐴)) = dom 𝐹)
97, 8mpbi 229 . . . . 5 (dom 𝐹 ∪ (𝐹𝐴)) = dom 𝐹
106, 9eqtri 2753 . . . 4 ((𝐹 “ ran 𝐹) ∪ (𝐹𝐴)) = dom 𝐹
114, 10eqtrdi 2781 . . 3 (𝐴 = (ran 𝐹𝐴) → (𝐹𝐴) = dom 𝐹)
1211eqcoms 2733 . 2 ((ran 𝐹𝐴) = 𝐴 → (𝐹𝐴) = dom 𝐹)
131, 12sylbi 216 1 (ran 𝐹𝐴 → (𝐹𝐴) = dom 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  cun 3939  wss 3941  ccnv 5672  dom cdm 5673  ran crn 5674  cima 5676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-br 5145  df-opab 5207  df-xp 5679  df-cnv 5681  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686
This theorem is referenced by:  fnco  6667  fimacnv  6739
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