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| Mirrors > Home > MPE Home > Th. List > f1imacnv | Structured version Visualization version GIF version | ||
| Description: Preimage of an image. (Contributed by NM, 30-Sep-2004.) | 
| Ref | Expression | 
|---|---|
| f1imacnv | ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (◡𝐹 “ (𝐹 “ 𝐶)) = 𝐶) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | resima 6032 | . 2 ⊢ ((◡𝐹 ↾ (𝐹 “ 𝐶)) “ (𝐹 “ 𝐶)) = (◡𝐹 “ (𝐹 “ 𝐶)) | |
| 2 | df-f1 6565 | . . . . . . 7 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹)) | |
| 3 | 2 | simprbi 496 | . . . . . 6 ⊢ (𝐹:𝐴–1-1→𝐵 → Fun ◡𝐹) | 
| 4 | 3 | adantr 480 | . . . . 5 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → Fun ◡𝐹) | 
| 5 | funcnvres 6643 | . . . . 5 ⊢ (Fun ◡𝐹 → ◡(𝐹 ↾ 𝐶) = (◡𝐹 ↾ (𝐹 “ 𝐶))) | |
| 6 | 4, 5 | syl 17 | . . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → ◡(𝐹 ↾ 𝐶) = (◡𝐹 ↾ (𝐹 “ 𝐶))) | 
| 7 | 6 | imaeq1d 6076 | . . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (◡(𝐹 ↾ 𝐶) “ (𝐹 “ 𝐶)) = ((◡𝐹 ↾ (𝐹 “ 𝐶)) “ (𝐹 “ 𝐶))) | 
| 8 | f1ores 6861 | . . . . 5 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶)) | |
| 9 | f1ocnv 6859 | . . . . 5 ⊢ ((𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶) → ◡(𝐹 ↾ 𝐶):(𝐹 “ 𝐶)–1-1-onto→𝐶) | |
| 10 | 8, 9 | syl 17 | . . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → ◡(𝐹 ↾ 𝐶):(𝐹 “ 𝐶)–1-1-onto→𝐶) | 
| 11 | imadmrn 6087 | . . . . 5 ⊢ (◡(𝐹 ↾ 𝐶) “ dom ◡(𝐹 ↾ 𝐶)) = ran ◡(𝐹 ↾ 𝐶) | |
| 12 | f1odm 6851 | . . . . . 6 ⊢ (◡(𝐹 ↾ 𝐶):(𝐹 “ 𝐶)–1-1-onto→𝐶 → dom ◡(𝐹 ↾ 𝐶) = (𝐹 “ 𝐶)) | |
| 13 | 12 | imaeq2d 6077 | . . . . 5 ⊢ (◡(𝐹 ↾ 𝐶):(𝐹 “ 𝐶)–1-1-onto→𝐶 → (◡(𝐹 ↾ 𝐶) “ dom ◡(𝐹 ↾ 𝐶)) = (◡(𝐹 ↾ 𝐶) “ (𝐹 “ 𝐶))) | 
| 14 | f1ofo 6854 | . . . . . 6 ⊢ (◡(𝐹 ↾ 𝐶):(𝐹 “ 𝐶)–1-1-onto→𝐶 → ◡(𝐹 ↾ 𝐶):(𝐹 “ 𝐶)–onto→𝐶) | |
| 15 | forn 6822 | . . . . . 6 ⊢ (◡(𝐹 ↾ 𝐶):(𝐹 “ 𝐶)–onto→𝐶 → ran ◡(𝐹 ↾ 𝐶) = 𝐶) | |
| 16 | 14, 15 | syl 17 | . . . . 5 ⊢ (◡(𝐹 ↾ 𝐶):(𝐹 “ 𝐶)–1-1-onto→𝐶 → ran ◡(𝐹 ↾ 𝐶) = 𝐶) | 
| 17 | 11, 13, 16 | 3eqtr3a 2800 | . . . 4 ⊢ (◡(𝐹 ↾ 𝐶):(𝐹 “ 𝐶)–1-1-onto→𝐶 → (◡(𝐹 ↾ 𝐶) “ (𝐹 “ 𝐶)) = 𝐶) | 
| 18 | 10, 17 | syl 17 | . . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (◡(𝐹 ↾ 𝐶) “ (𝐹 “ 𝐶)) = 𝐶) | 
| 19 | 7, 18 | eqtr3d 2778 | . 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → ((◡𝐹 ↾ (𝐹 “ 𝐶)) “ (𝐹 “ 𝐶)) = 𝐶) | 
| 20 | 1, 19 | eqtr3id 2790 | 1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (◡𝐹 “ (𝐹 “ 𝐶)) = 𝐶) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ⊆ wss 3950 ◡ccnv 5683 dom cdm 5684 ran crn 5685 ↾ cres 5686 “ cima 5687 Fun wfun 6554 ⟶wf 6556 –1-1→wf1 6557 –onto→wfo 6558 –1-1-onto→wf1o 6559 | 
| 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-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-sb 2064 df-mo 2539 df-eu 2568 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-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 | 
| This theorem is referenced by: f1opw2 7689 mptcnfimad 8012 ssenen 9192 f1opwfi 9397 isf34lem3 10416 subggim 19285 gicsubgen 19298 cnt1 23359 basqtop 23720 tgqtop 23721 hmeoopn 23775 hmeocld 23776 hmeontr 23778 qtopf1 23825 f1otrg 28880 tpr2rico 33912 eulerpartlemmf 34378 ballotlemscr 34522 ballotlemrinv0 34536 cvmlift2lem9a 35309 grpokerinj 37901 uspgrimprop 47878 grimcnv 47884 isubgr3stgrlem8 47945 | 
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