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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 5935 for subsets. (Contributed by AV, 18-Sep-2024.) |
Ref | Expression |
---|---|
cnvimassrndm | ⊢ (ran 𝐹 ⊆ 𝐴 → (◡𝐹 “ 𝐴) = dom 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssequn1 4080 | . 2 ⊢ (ran 𝐹 ⊆ 𝐴 ↔ (ran 𝐹 ∪ 𝐴) = 𝐴) | |
2 | imaeq2 5910 | . . . . 5 ⊢ (𝐴 = (ran 𝐹 ∪ 𝐴) → (◡𝐹 “ 𝐴) = (◡𝐹 “ (ran 𝐹 ∪ 𝐴))) | |
3 | imaundi 5993 | . . . . 5 ⊢ (◡𝐹 “ (ran 𝐹 ∪ 𝐴)) = ((◡𝐹 “ ran 𝐹) ∪ (◡𝐹 “ 𝐴)) | |
4 | 2, 3 | eqtrdi 2787 | . . . 4 ⊢ (𝐴 = (ran 𝐹 ∪ 𝐴) → (◡𝐹 “ 𝐴) = ((◡𝐹 “ ran 𝐹) ∪ (◡𝐹 “ 𝐴))) |
5 | cnvimarndm 5935 | . . . . . 6 ⊢ (◡𝐹 “ ran 𝐹) = dom 𝐹 | |
6 | 5 | uneq1i 4059 | . . . . 5 ⊢ ((◡𝐹 “ ran 𝐹) ∪ (◡𝐹 “ 𝐴)) = (dom 𝐹 ∪ (◡𝐹 “ 𝐴)) |
7 | cnvimass 5934 | . . . . . 6 ⊢ (◡𝐹 “ 𝐴) ⊆ dom 𝐹 | |
8 | ssequn2 4083 | . . . . . 6 ⊢ ((◡𝐹 “ 𝐴) ⊆ dom 𝐹 ↔ (dom 𝐹 ∪ (◡𝐹 “ 𝐴)) = dom 𝐹) | |
9 | 7, 8 | mpbi 233 | . . . . 5 ⊢ (dom 𝐹 ∪ (◡𝐹 “ 𝐴)) = dom 𝐹 |
10 | 6, 9 | eqtri 2759 | . . . 4 ⊢ ((◡𝐹 “ ran 𝐹) ∪ (◡𝐹 “ 𝐴)) = dom 𝐹 |
11 | 4, 10 | eqtrdi 2787 | . . 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 220 | 1 ⊢ (ran 𝐹 ⊆ 𝐴 → (◡𝐹 “ 𝐴) = dom 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∪ cun 3851 ⊆ wss 3853 ◡ccnv 5535 dom cdm 5536 ran crn 5537 “ cima 5539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-xp 5542 df-cnv 5544 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 |
This theorem is referenced by: fnco 6472 fimacnv 6545 |
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