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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 6100 for subsets. (Contributed by AV, 18-Sep-2024.) | 
| Ref | Expression | 
|---|---|
| cnvimassrndm | ⊢ (ran 𝐹 ⊆ 𝐴 → (◡𝐹 “ 𝐴) = dom 𝐹) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ssequn1 4185 | . 2 ⊢ (ran 𝐹 ⊆ 𝐴 ↔ (ran 𝐹 ∪ 𝐴) = 𝐴) | |
| 2 | imaeq2 6073 | . . . . 5 ⊢ (𝐴 = (ran 𝐹 ∪ 𝐴) → (◡𝐹 “ 𝐴) = (◡𝐹 “ (ran 𝐹 ∪ 𝐴))) | |
| 3 | imaundi 6168 | . . . . 5 ⊢ (◡𝐹 “ (ran 𝐹 ∪ 𝐴)) = ((◡𝐹 “ ran 𝐹) ∪ (◡𝐹 “ 𝐴)) | |
| 4 | 2, 3 | eqtrdi 2792 | . . . 4 ⊢ (𝐴 = (ran 𝐹 ∪ 𝐴) → (◡𝐹 “ 𝐴) = ((◡𝐹 “ ran 𝐹) ∪ (◡𝐹 “ 𝐴))) | 
| 5 | cnvimarndm 6100 | . . . . . 6 ⊢ (◡𝐹 “ ran 𝐹) = dom 𝐹 | |
| 6 | 5 | uneq1i 4163 | . . . . 5 ⊢ ((◡𝐹 “ ran 𝐹) ∪ (◡𝐹 “ 𝐴)) = (dom 𝐹 ∪ (◡𝐹 “ 𝐴)) | 
| 7 | cnvimass 6099 | . . . . . 6 ⊢ (◡𝐹 “ 𝐴) ⊆ dom 𝐹 | |
| 8 | ssequn2 4188 | . . . . . 6 ⊢ ((◡𝐹 “ 𝐴) ⊆ dom 𝐹 ↔ (dom 𝐹 ∪ (◡𝐹 “ 𝐴)) = dom 𝐹) | |
| 9 | 7, 8 | mpbi 230 | . . . . 5 ⊢ (dom 𝐹 ∪ (◡𝐹 “ 𝐴)) = dom 𝐹 | 
| 10 | 6, 9 | eqtri 2764 | . . . 4 ⊢ ((◡𝐹 “ ran 𝐹) ∪ (◡𝐹 “ 𝐴)) = dom 𝐹 | 
| 11 | 4, 10 | eqtrdi 2792 | . . 3 ⊢ (𝐴 = (ran 𝐹 ∪ 𝐴) → (◡𝐹 “ 𝐴) = dom 𝐹) | 
| 12 | 11 | eqcoms 2744 | . 2 ⊢ ((ran 𝐹 ∪ 𝐴) = 𝐴 → (◡𝐹 “ 𝐴) = dom 𝐹) | 
| 13 | 1, 12 | sylbi 217 | 1 ⊢ (ran 𝐹 ⊆ 𝐴 → (◡𝐹 “ 𝐴) = 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 | 
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