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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 6760 | . . . . 5 ⊢ (𝐺:𝐴–onto→𝐵 → ran 𝐺 = 𝐵) | |
2 | 1 | eqcomd 2743 | . . . 4 ⊢ (𝐺:𝐴–onto→𝐵 → 𝐵 = ran 𝐺) |
3 | 2 | imaeq2d 6014 | . . 3 ⊢ (𝐺:𝐴–onto→𝐵 → (◡𝐺 “ 𝐵) = (◡𝐺 “ ran 𝐺)) |
4 | cnvimarndm 6035 | . . 3 ⊢ (◡𝐺 “ ran 𝐺) = dom 𝐺 | |
5 | 3, 4 | eqtrdi 2793 | . 2 ⊢ (𝐺:𝐴–onto→𝐵 → (◡𝐺 “ 𝐵) = dom 𝐺) |
6 | fof 6757 | . . 3 ⊢ (𝐺:𝐴–onto→𝐵 → 𝐺:𝐴⟶𝐵) | |
7 | 6 | fdmd 6680 | . 2 ⊢ (𝐺:𝐴–onto→𝐵 → dom 𝐺 = 𝐴) |
8 | 5, 7 | eqtrd 2777 | 1 ⊢ (𝐺:𝐴–onto→𝐵 → (◡𝐺 “ 𝐵) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ◡ccnv 5633 dom cdm 5634 ran crn 5635 “ cima 5637 –onto→wfo 6495 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-xp 5640 df-cnv 5642 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-fn 6500 df-f 6501 df-fo 6503 |
This theorem is referenced by: foco 6771 |
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