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Mirrors > Home > MPE Home > Th. List > fnbrfvb | Structured version Visualization version GIF version |
Description: Equivalence of function value and binary relation. (Contributed by NM, 19-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
fnbrfvb | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = 𝐶 ↔ 𝐵𝐹𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2651 | . . . 4 ⊢ (𝐹‘𝐵) = (𝐹‘𝐵) | |
2 | fvex 6239 | . . . . 5 ⊢ (𝐹‘𝐵) ∈ V | |
3 | eqeq2 2662 | . . . . . . 7 ⊢ (𝑥 = (𝐹‘𝐵) → ((𝐹‘𝐵) = 𝑥 ↔ (𝐹‘𝐵) = (𝐹‘𝐵))) | |
4 | breq2 4689 | . . . . . . 7 ⊢ (𝑥 = (𝐹‘𝐵) → (𝐵𝐹𝑥 ↔ 𝐵𝐹(𝐹‘𝐵))) | |
5 | 3, 4 | bibi12d 334 | . . . . . 6 ⊢ (𝑥 = (𝐹‘𝐵) → (((𝐹‘𝐵) = 𝑥 ↔ 𝐵𝐹𝑥) ↔ ((𝐹‘𝐵) = (𝐹‘𝐵) ↔ 𝐵𝐹(𝐹‘𝐵)))) |
6 | 5 | imbi2d 329 | . . . . 5 ⊢ (𝑥 = (𝐹‘𝐵) → (((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = 𝑥 ↔ 𝐵𝐹𝑥)) ↔ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = (𝐹‘𝐵) ↔ 𝐵𝐹(𝐹‘𝐵))))) |
7 | fneu 6033 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ∃!𝑥 𝐵𝐹𝑥) | |
8 | tz6.12c 6251 | . . . . . 6 ⊢ (∃!𝑥 𝐵𝐹𝑥 → ((𝐹‘𝐵) = 𝑥 ↔ 𝐵𝐹𝑥)) | |
9 | 7, 8 | syl 17 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = 𝑥 ↔ 𝐵𝐹𝑥)) |
10 | 2, 6, 9 | vtocl 3290 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = (𝐹‘𝐵) ↔ 𝐵𝐹(𝐹‘𝐵))) |
11 | 1, 10 | mpbii 223 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵𝐹(𝐹‘𝐵)) |
12 | breq2 4689 | . . 3 ⊢ ((𝐹‘𝐵) = 𝐶 → (𝐵𝐹(𝐹‘𝐵) ↔ 𝐵𝐹𝐶)) | |
13 | 11, 12 | syl5ibcom 235 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = 𝐶 → 𝐵𝐹𝐶)) |
14 | fnfun 6026 | . . . 4 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹) | |
15 | funbrfv 6272 | . . . 4 ⊢ (Fun 𝐹 → (𝐵𝐹𝐶 → (𝐹‘𝐵) = 𝐶)) | |
16 | 14, 15 | syl 17 | . . 3 ⊢ (𝐹 Fn 𝐴 → (𝐵𝐹𝐶 → (𝐹‘𝐵) = 𝐶)) |
17 | 16 | adantr 480 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐵𝐹𝐶 → (𝐹‘𝐵) = 𝐶)) |
18 | 13, 17 | impbid 202 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = 𝐶 ↔ 𝐵𝐹𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ∃!weu 2498 class class class wbr 4685 Fun wfun 5920 Fn wfn 5921 ‘cfv 5926 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-iota 5889 df-fun 5928 df-fn 5929 df-fv 5934 |
This theorem is referenced by: fnopfvb 6275 funbrfvb 6276 fnbrfvb2 6278 dffn5 6280 feqmptdf 6290 fnsnfv 6297 fndmdif 6361 dffo4 6415 dff13 6552 isomin 6627 isoini 6628 br1steqg 7232 br2ndeqg 7233 1stconst 7310 2ndconst 7311 fsplit 7327 seqomlem3 7592 seqomlem4 7593 nqerrel 9792 imasleval 16248 znleval 19951 axcontlem5 25893 elnlfn 28915 adjbd1o 29072 fcoinvbr 29545 elintfv 31788 fv1stcnv 31804 fv2ndcnv 31805 trpredpred 31852 scutun12 32042 madeval2 32061 fvbigcup 32134 fvsingle 32152 imageval 32162 brfullfun 32180 bj-mptval 33195 unccur 33522 poimirlem2 33541 poimirlem23 33562 pw2f1ocnv 37921 brcoffn 38645 funressnfv 41529 fnbrafvb 41555 |
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