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Mirrors > Home > NFE Home > Th. List > f1imacnv | GIF version |
Description: Preimage of an image. (Contributed by set.mm contributors, 30-Sep-2004.) |
Ref | Expression |
---|---|
f1imacnv | ⊢ ((F:A–1-1→B ∧ C ⊆ A) → (◡F “ (F “ C)) = C) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resima 5006 | . 2 ⊢ ((◡F ↾ (F “ C)) “ (F “ C)) = (◡F “ (F “ C)) | |
2 | df-f1 4792 | . . . . . 6 ⊢ (F:A–1-1→B ↔ (F:A–→B ∧ Fun ◡F)) | |
3 | 2 | simprbi 450 | . . . . 5 ⊢ (F:A–1-1→B → Fun ◡F) |
4 | 3 | adantr 451 | . . . 4 ⊢ ((F:A–1-1→B ∧ C ⊆ A) → Fun ◡F) |
5 | funcnvres 5165 | . . . 4 ⊢ (Fun ◡F → ◡(F ↾ C) = (◡F ↾ (F “ C))) | |
6 | imaeq1 4937 | . . . 4 ⊢ (◡(F ↾ C) = (◡F ↾ (F “ C)) → (◡(F ↾ C) “ (F “ C)) = ((◡F ↾ (F “ C)) “ (F “ C))) | |
7 | 4, 5, 6 | 3syl 18 | . . 3 ⊢ ((F:A–1-1→B ∧ C ⊆ A) → (◡(F ↾ C) “ (F “ C)) = ((◡F ↾ (F “ C)) “ (F “ C))) |
8 | f1ores 5300 | . . . 4 ⊢ ((F:A–1-1→B ∧ C ⊆ A) → (F ↾ C):C–1-1-onto→(F “ C)) | |
9 | f1ocnv 5299 | . . . 4 ⊢ ((F ↾ C):C–1-1-onto→(F “ C) → ◡(F ↾ C):(F “ C)–1-1-onto→C) | |
10 | imadmrn 5008 | . . . . . 6 ⊢ (◡(F ↾ C) “ dom ◡(F ↾ C)) = ran ◡(F ↾ C) | |
11 | f1of 5287 | . . . . . . 7 ⊢ (◡(F ↾ C):(F “ C)–1-1-onto→C → ◡(F ↾ C):(F “ C)–→C) | |
12 | fdm 5226 | . . . . . . 7 ⊢ (◡(F ↾ C):(F “ C)–→C → dom ◡(F ↾ C) = (F “ C)) | |
13 | imaeq2 4938 | . . . . . . 7 ⊢ (dom ◡(F ↾ C) = (F “ C) → (◡(F ↾ C) “ dom ◡(F ↾ C)) = (◡(F ↾ C) “ (F “ C))) | |
14 | 11, 12, 13 | 3syl 18 | . . . . . 6 ⊢ (◡(F ↾ C):(F “ C)–1-1-onto→C → (◡(F ↾ C) “ dom ◡(F ↾ C)) = (◡(F ↾ C) “ (F “ C))) |
15 | 10, 14 | syl5reqr 2400 | . . . . 5 ⊢ (◡(F ↾ C):(F “ C)–1-1-onto→C → (◡(F ↾ C) “ (F “ C)) = ran ◡(F ↾ C)) |
16 | f1ofo 5293 | . . . . . 6 ⊢ (◡(F ↾ C):(F “ C)–1-1-onto→C → ◡(F ↾ C):(F “ C)–onto→C) | |
17 | forn 5272 | . . . . . 6 ⊢ (◡(F ↾ C):(F “ C)–onto→C → ran ◡(F ↾ C) = C) | |
18 | 16, 17 | syl 15 | . . . . 5 ⊢ (◡(F ↾ C):(F “ C)–1-1-onto→C → ran ◡(F ↾ C) = C) |
19 | 15, 18 | eqtrd 2385 | . . . 4 ⊢ (◡(F ↾ C):(F “ C)–1-1-onto→C → (◡(F ↾ C) “ (F “ C)) = C) |
20 | 8, 9, 19 | 3syl 18 | . . 3 ⊢ ((F:A–1-1→B ∧ C ⊆ A) → (◡(F ↾ C) “ (F “ C)) = C) |
21 | 7, 20 | eqtr3d 2387 | . 2 ⊢ ((F:A–1-1→B ∧ C ⊆ A) → ((◡F ↾ (F “ C)) “ (F “ C)) = C) |
22 | 1, 21 | syl5eqr 2399 | 1 ⊢ ((F:A–1-1→B ∧ C ⊆ A) → (◡F “ (F “ C)) = C) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 = wceq 1642 ⊆ wss 3257 “ cima 4722 ◡ccnv 4771 dom cdm 4772 ran crn 4773 ↾ cres 4774 Fun wfun 4775 –→wf 4777 –1-1→wf1 4778 –onto→wfo 4779 –1-1-onto→wf1o 4780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-co 4726 df-ima 4727 df-id 4767 df-xp 4784 df-cnv 4785 df-rn 4786 df-dm 4787 df-res 4788 df-fun 4789 df-fn 4790 df-f 4791 df-f1 4792 df-fo 4793 df-f1o 4794 |
This theorem is referenced by: (None) |
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