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Theorem cnvimassrndm 6150
Description: The preimage of a superset of the range of a class is the domain of the class. Generalization of cnvimarndm 6080 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 4176 . 2 (ran 𝐹𝐴 ↔ (ran 𝐹𝐴) = 𝐴)
2 imaeq2 6053 . . . . 5 (𝐴 = (ran 𝐹𝐴) → (𝐹𝐴) = (𝐹 “ (ran 𝐹𝐴)))
3 imaundi 6148 . . . . 5 (𝐹 “ (ran 𝐹𝐴)) = ((𝐹 “ ran 𝐹) ∪ (𝐹𝐴))
42, 3eqtrdi 2783 . . . 4 (𝐴 = (ran 𝐹𝐴) → (𝐹𝐴) = ((𝐹 “ ran 𝐹) ∪ (𝐹𝐴)))
5 cnvimarndm 6080 . . . . . 6 (𝐹 “ ran 𝐹) = dom 𝐹
65uneq1i 4155 . . . . 5 ((𝐹 “ ran 𝐹) ∪ (𝐹𝐴)) = (dom 𝐹 ∪ (𝐹𝐴))
7 cnvimass 6079 . . . . . 6 (𝐹𝐴) ⊆ dom 𝐹
8 ssequn2 4179 . . . . . 6 ((𝐹𝐴) ⊆ dom 𝐹 ↔ (dom 𝐹 ∪ (𝐹𝐴)) = dom 𝐹)
97, 8mpbi 229 . . . . 5 (dom 𝐹 ∪ (𝐹𝐴)) = dom 𝐹
106, 9eqtri 2755 . . . 4 ((𝐹 “ ran 𝐹) ∪ (𝐹𝐴)) = dom 𝐹
114, 10eqtrdi 2783 . . 3 (𝐴 = (ran 𝐹𝐴) → (𝐹𝐴) = dom 𝐹)
1211eqcoms 2735 . 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 1534  cun 3942  wss 3944  ccnv 5671  dom cdm 5672  ran crn 5673  cima 5675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-xp 5678  df-cnv 5680  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685
This theorem is referenced by:  fnco  6666  fimacnv  6739
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