MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnvimassrndm Structured version   Visualization version   GIF version

Theorem cnvimassrndm 6171
Description: The preimage of a superset of the range of a class is the domain of the class. Generalization of cnvimarndm 6100 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 4185 . 2 (ran 𝐹𝐴 ↔ (ran 𝐹𝐴) = 𝐴)
2 imaeq2 6073 . . . . 5 (𝐴 = (ran 𝐹𝐴) → (𝐹𝐴) = (𝐹 “ (ran 𝐹𝐴)))
3 imaundi 6168 . . . . 5 (𝐹 “ (ran 𝐹𝐴)) = ((𝐹 “ ran 𝐹) ∪ (𝐹𝐴))
42, 3eqtrdi 2792 . . . 4 (𝐴 = (ran 𝐹𝐴) → (𝐹𝐴) = ((𝐹 “ ran 𝐹) ∪ (𝐹𝐴)))
5 cnvimarndm 6100 . . . . . 6 (𝐹 “ ran 𝐹) = dom 𝐹
65uneq1i 4163 . . . . 5 ((𝐹 “ ran 𝐹) ∪ (𝐹𝐴)) = (dom 𝐹 ∪ (𝐹𝐴))
7 cnvimass 6099 . . . . . 6 (𝐹𝐴) ⊆ dom 𝐹
8 ssequn2 4188 . . . . . 6 ((𝐹𝐴) ⊆ dom 𝐹 ↔ (dom 𝐹 ∪ (𝐹𝐴)) = dom 𝐹)
97, 8mpbi 230 . . . . 5 (dom 𝐹 ∪ (𝐹𝐴)) = dom 𝐹
106, 9eqtri 2764 . . . 4 ((𝐹 “ ran 𝐹) ∪ (𝐹𝐴)) = dom 𝐹
114, 10eqtrdi 2792 . . 3 (𝐴 = (ran 𝐹𝐴) → (𝐹𝐴) = dom 𝐹)
1211eqcoms 2744 . 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 1539  cun 3948  wss 3950  ccnv 5683  dom cdm 5684  ran crn 5685  cima 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697
This theorem is referenced by:  fnco  6685  fimacnv  6757
  Copyright terms: Public domain W3C validator