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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 2795 | . . . 4 ⊢ (𝐹‘𝐵) = (𝐹‘𝐵) | |
2 | fvex 6551 | . . . . 5 ⊢ (𝐹‘𝐵) ∈ V | |
3 | eqeq2 2806 | . . . . . . 7 ⊢ (𝑥 = (𝐹‘𝐵) → ((𝐹‘𝐵) = 𝑥 ↔ (𝐹‘𝐵) = (𝐹‘𝐵))) | |
4 | breq2 4966 | . . . . . . 7 ⊢ (𝑥 = (𝐹‘𝐵) → (𝐵𝐹𝑥 ↔ 𝐵𝐹(𝐹‘𝐵))) | |
5 | 3, 4 | bibi12d 347 | . . . . . 6 ⊢ (𝑥 = (𝐹‘𝐵) → (((𝐹‘𝐵) = 𝑥 ↔ 𝐵𝐹𝑥) ↔ ((𝐹‘𝐵) = (𝐹‘𝐵) ↔ 𝐵𝐹(𝐹‘𝐵)))) |
6 | 5 | imbi2d 342 | . . . . 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 6331 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ∃!𝑥 𝐵𝐹𝑥) | |
8 | tz6.12c 6563 | . . . . . 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 3502 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = (𝐹‘𝐵) ↔ 𝐵𝐹(𝐹‘𝐵))) |
11 | 1, 10 | mpbii 234 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵𝐹(𝐹‘𝐵)) |
12 | breq2 4966 | . . 3 ⊢ ((𝐹‘𝐵) = 𝐶 → (𝐵𝐹(𝐹‘𝐵) ↔ 𝐵𝐹𝐶)) | |
13 | 11, 12 | syl5ibcom 246 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = 𝐶 → 𝐵𝐹𝐶)) |
14 | fnfun 6323 | . . . 4 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹) | |
15 | funbrfv 6584 | . . . 4 ⊢ (Fun 𝐹 → (𝐵𝐹𝐶 → (𝐹‘𝐵) = 𝐶)) | |
16 | 14, 15 | syl 17 | . . 3 ⊢ (𝐹 Fn 𝐴 → (𝐵𝐹𝐶 → (𝐹‘𝐵) = 𝐶)) |
17 | 16 | adantr 481 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐵𝐹𝐶 → (𝐹‘𝐵) = 𝐶)) |
18 | 13, 17 | impbid 213 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = 𝐶 ↔ 𝐵𝐹𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1522 ∈ wcel 2081 ∃!weu 2611 class class class wbr 4962 Fun wfun 6219 Fn wfn 6220 ‘cfv 6225 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pr 5221 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-sbc 3707 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-br 4963 df-opab 5025 df-id 5348 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-iota 6189 df-fun 6227 df-fn 6228 df-fv 6233 |
This theorem is referenced by: fnopfvb 6587 funbrfvb 6588 fnbrfvb2 6590 dffn5 6592 feqmptdf 6603 fnsnfv 6610 fndmdif 6677 dffo4 6732 dff13 6878 isomin 6953 isoini 6954 br1steqg 7567 br2ndeqg 7568 1stconst 7651 2ndconst 7652 fsplit 7668 seqomlem3 7939 seqomlem4 7940 nqerrel 10200 imasleval 16643 znleval 20383 axcontlem5 26437 elnlfn 29396 adjbd1o 29553 fcoinvbr 30048 fv1stcnv 32628 fv2ndcnv 32629 trpredpred 32676 scutun12 32880 madeval2 32899 fvbigcup 32972 fvsingle 32990 imageval 33000 brfullfun 33018 bj-mptval 34007 unccur 34406 poimirlem2 34425 poimirlem23 34446 pw2f1ocnv 39119 brcoffn 39865 funressnfv 42794 fnbrafvb 42869 |
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