Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6614 | . . . . 5 ⊢ (𝐺:𝐴–onto→𝐵 → ran 𝐺 = 𝐵) | |
2 | 1 | eqcomd 2742 | . . . 4 ⊢ (𝐺:𝐴–onto→𝐵 → 𝐵 = ran 𝐺) |
3 | 2 | imaeq2d 5914 | . . 3 ⊢ (𝐺:𝐴–onto→𝐵 → (◡𝐺 “ 𝐵) = (◡𝐺 “ ran 𝐺)) |
4 | cnvimarndm 5935 | . . 3 ⊢ (◡𝐺 “ ran 𝐺) = dom 𝐺 | |
5 | 3, 4 | eqtrdi 2787 | . 2 ⊢ (𝐺:𝐴–onto→𝐵 → (◡𝐺 “ 𝐵) = dom 𝐺) |
6 | fof 6611 | . . 3 ⊢ (𝐺:𝐴–onto→𝐵 → 𝐺:𝐴⟶𝐵) | |
7 | 6 | fdmd 6534 | . 2 ⊢ (𝐺:𝐴–onto→𝐵 → dom 𝐺 = 𝐴) |
8 | 5, 7 | eqtrd 2771 | 1 ⊢ (𝐺:𝐴–onto→𝐵 → (◡𝐺 “ 𝐵) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ◡ccnv 5535 dom cdm 5536 ran crn 5537 “ cima 5539 –onto→wfo 6356 |
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-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-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 df-fn 6361 df-f 6362 df-fo 6364 |
This theorem is referenced by: foco 6625 |
Copyright terms: Public domain | W3C validator |