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Theorem cnvimassrndm 6174
Description: The preimage of a superset of the range of a class is the domain of the class. Generalization of cnvimarndm 6103 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 4196 . 2 (ran 𝐹𝐴 ↔ (ran 𝐹𝐴) = 𝐴)
2 imaeq2 6076 . . . . 5 (𝐴 = (ran 𝐹𝐴) → (𝐹𝐴) = (𝐹 “ (ran 𝐹𝐴)))
3 imaundi 6172 . . . . 5 (𝐹 “ (ran 𝐹𝐴)) = ((𝐹 “ ran 𝐹) ∪ (𝐹𝐴))
42, 3eqtrdi 2791 . . . 4 (𝐴 = (ran 𝐹𝐴) → (𝐹𝐴) = ((𝐹 “ ran 𝐹) ∪ (𝐹𝐴)))
5 cnvimarndm 6103 . . . . . 6 (𝐹 “ ran 𝐹) = dom 𝐹
65uneq1i 4174 . . . . 5 ((𝐹 “ ran 𝐹) ∪ (𝐹𝐴)) = (dom 𝐹 ∪ (𝐹𝐴))
7 cnvimass 6102 . . . . . 6 (𝐹𝐴) ⊆ dom 𝐹
8 ssequn2 4199 . . . . . 6 ((𝐹𝐴) ⊆ dom 𝐹 ↔ (dom 𝐹 ∪ (𝐹𝐴)) = dom 𝐹)
97, 8mpbi 230 . . . . 5 (dom 𝐹 ∪ (𝐹𝐴)) = dom 𝐹
106, 9eqtri 2763 . . . 4 ((𝐹 “ ran 𝐹) ∪ (𝐹𝐴)) = dom 𝐹
114, 10eqtrdi 2791 . . 3 (𝐴 = (ran 𝐹𝐴) → (𝐹𝐴) = dom 𝐹)
1211eqcoms 2743 . 2 ((ran 𝐹𝐴) = 𝐴 → (𝐹𝐴) = dom 𝐹)
131, 12sylbi 217 1 (ran 𝐹𝐴 → (𝐹𝐴) = dom 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  cun 3961  wss 3963  ccnv 5688  dom cdm 5689  ran crn 5690  cima 5692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702
This theorem is referenced by:  fnco  6687  fimacnv  6759
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