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Mirrors > Home > NFE Home > Th. List > elpreima | GIF version |
Description: Membership in the preimage of a set under a function. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
elpreima | ⊢ (F Fn A → (B ∈ (◡F “ C) ↔ (B ∈ A ∧ (F ‘B) ∈ C))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvimass 5016 | . . . . 5 ⊢ (◡F “ C) ⊆ dom F | |
2 | 1 | sseli 3269 | . . . 4 ⊢ (B ∈ (◡F “ C) → B ∈ dom F) |
3 | fndm 5182 | . . . . 5 ⊢ (F Fn A → dom F = A) | |
4 | 3 | eleq2d 2420 | . . . 4 ⊢ (F Fn A → (B ∈ dom F ↔ B ∈ A)) |
5 | 2, 4 | syl5ib 210 | . . 3 ⊢ (F Fn A → (B ∈ (◡F “ C) → B ∈ A)) |
6 | fnfun 5181 | . . . . 5 ⊢ (F Fn A → Fun F) | |
7 | fvimacnvi 5402 | . . . . 5 ⊢ ((Fun F ∧ B ∈ (◡F “ C)) → (F ‘B) ∈ C) | |
8 | 6, 7 | sylan 457 | . . . 4 ⊢ ((F Fn A ∧ B ∈ (◡F “ C)) → (F ‘B) ∈ C) |
9 | 8 | ex 423 | . . 3 ⊢ (F Fn A → (B ∈ (◡F “ C) → (F ‘B) ∈ C)) |
10 | 5, 9 | jcad 519 | . 2 ⊢ (F Fn A → (B ∈ (◡F “ C) → (B ∈ A ∧ (F ‘B) ∈ C))) |
11 | fvimacnv 5403 | . . . . 5 ⊢ ((Fun F ∧ B ∈ dom F) → ((F ‘B) ∈ C ↔ B ∈ (◡F “ C))) | |
12 | 11 | funfni 5183 | . . . 4 ⊢ ((F Fn A ∧ B ∈ A) → ((F ‘B) ∈ C ↔ B ∈ (◡F “ C))) |
13 | 12 | biimpd 198 | . . 3 ⊢ ((F Fn A ∧ B ∈ A) → ((F ‘B) ∈ C → B ∈ (◡F “ C))) |
14 | 13 | expimpd 586 | . 2 ⊢ (F Fn A → ((B ∈ A ∧ (F ‘B) ∈ C) → B ∈ (◡F “ C))) |
15 | 10, 14 | impbid 183 | 1 ⊢ (F Fn A → (B ∈ (◡F “ C) ↔ (B ∈ A ∧ (F ‘B) ∈ C))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 ∈ wcel 1710 “ cima 4722 ◡ccnv 4771 dom cdm 4772 Fun wfun 4775 Fn wfn 4776 ‘cfv 4781 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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-fv 4795 |
This theorem is referenced by: unpreima 5408 inpreima 5409 respreima 5410 |
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