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Mirrors > Home > MPE Home > Th. List > cnvimassrndm | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
cnvimassrndm | ⊢ (ran 𝐹 ⊆ 𝐴 → (◡𝐹 “ 𝐴) = dom 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssequn1 4175 | . 2 ⊢ (ran 𝐹 ⊆ 𝐴 ↔ (ran 𝐹 ∪ 𝐴) = 𝐴) | |
2 | imaeq2 6055 | . . . . 5 ⊢ (𝐴 = (ran 𝐹 ∪ 𝐴) → (◡𝐹 “ 𝐴) = (◡𝐹 “ (ran 𝐹 ∪ 𝐴))) | |
3 | imaundi 6150 | . . . . 5 ⊢ (◡𝐹 “ (ran 𝐹 ∪ 𝐴)) = ((◡𝐹 “ ran 𝐹) ∪ (◡𝐹 “ 𝐴)) | |
4 | 2, 3 | eqtrdi 2781 | . . . 4 ⊢ (𝐴 = (ran 𝐹 ∪ 𝐴) → (◡𝐹 “ 𝐴) = ((◡𝐹 “ ran 𝐹) ∪ (◡𝐹 “ 𝐴))) |
5 | cnvimarndm 6082 | . . . . . 6 ⊢ (◡𝐹 “ ran 𝐹) = dom 𝐹 | |
6 | 5 | uneq1i 4153 | . . . . 5 ⊢ ((◡𝐹 “ ran 𝐹) ∪ (◡𝐹 “ 𝐴)) = (dom 𝐹 ∪ (◡𝐹 “ 𝐴)) |
7 | cnvimass 6081 | . . . . . 6 ⊢ (◡𝐹 “ 𝐴) ⊆ dom 𝐹 | |
8 | ssequn2 4178 | . . . . . 6 ⊢ ((◡𝐹 “ 𝐴) ⊆ dom 𝐹 ↔ (dom 𝐹 ∪ (◡𝐹 “ 𝐴)) = dom 𝐹) | |
9 | 7, 8 | mpbi 229 | . . . . 5 ⊢ (dom 𝐹 ∪ (◡𝐹 “ 𝐴)) = dom 𝐹 |
10 | 6, 9 | eqtri 2753 | . . . 4 ⊢ ((◡𝐹 “ ran 𝐹) ∪ (◡𝐹 “ 𝐴)) = dom 𝐹 |
11 | 4, 10 | eqtrdi 2781 | . . 3 ⊢ (𝐴 = (ran 𝐹 ∪ 𝐴) → (◡𝐹 “ 𝐴) = dom 𝐹) |
12 | 11 | eqcoms 2733 | . 2 ⊢ ((ran 𝐹 ∪ 𝐴) = 𝐴 → (◡𝐹 “ 𝐴) = dom 𝐹) |
13 | 1, 12 | sylbi 216 | 1 ⊢ (ran 𝐹 ⊆ 𝐴 → (◡𝐹 “ 𝐴) = 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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