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Mirrors > Home > MPE Home > Th. List > focnvimacdmdm | Structured version Visualization version GIF version |
Description: The preimage of the codomain of a surjection is its domain. (Contributed by AV, 29-Sep-2024.) |
Ref | Expression |
---|---|
focnvimacdmdm | ⊢ (𝐺:𝐴–onto→𝐵 → (◡𝐺 “ 𝐵) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | forn 6807 | . . . . 5 ⊢ (𝐺:𝐴–onto→𝐵 → ran 𝐺 = 𝐵) | |
2 | 1 | eqcomd 2731 | . . . 4 ⊢ (𝐺:𝐴–onto→𝐵 → 𝐵 = ran 𝐺) |
3 | 2 | imaeq2d 6059 | . . 3 ⊢ (𝐺:𝐴–onto→𝐵 → (◡𝐺 “ 𝐵) = (◡𝐺 “ ran 𝐺)) |
4 | cnvimarndm 6082 | . . 3 ⊢ (◡𝐺 “ ran 𝐺) = dom 𝐺 | |
5 | 3, 4 | eqtrdi 2781 | . 2 ⊢ (𝐺:𝐴–onto→𝐵 → (◡𝐺 “ 𝐵) = dom 𝐺) |
6 | fof 6804 | . . 3 ⊢ (𝐺:𝐴–onto→𝐵 → 𝐺:𝐴⟶𝐵) | |
7 | 6 | fdmd 6727 | . 2 ⊢ (𝐺:𝐴–onto→𝐵 → dom 𝐺 = 𝐴) |
8 | 5, 7 | eqtrd 2765 | 1 ⊢ (𝐺:𝐴–onto→𝐵 → (◡𝐺 “ 𝐵) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ◡ccnv 5672 dom cdm 5673 ran crn 5674 “ cima 5676 –onto→wfo 6541 |
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-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-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 df-fn 6546 df-f 6547 df-fo 6549 |
This theorem is referenced by: foco 6818 |
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