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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 2737 | . . . 4 ⊢ (𝐹‘𝐵) = (𝐹‘𝐵) | |
| 2 | fvex 6919 | . . . . 5 ⊢ (𝐹‘𝐵) ∈ V | |
| 3 | eqeq2 2749 | . . . . . . 7 ⊢ (𝑥 = (𝐹‘𝐵) → ((𝐹‘𝐵) = 𝑥 ↔ (𝐹‘𝐵) = (𝐹‘𝐵))) | |
| 4 | breq2 5147 | . . . . . . 7 ⊢ (𝑥 = (𝐹‘𝐵) → (𝐵𝐹𝑥 ↔ 𝐵𝐹(𝐹‘𝐵))) | |
| 5 | 3, 4 | bibi12d 345 | . . . . . 6 ⊢ (𝑥 = (𝐹‘𝐵) → (((𝐹‘𝐵) = 𝑥 ↔ 𝐵𝐹𝑥) ↔ ((𝐹‘𝐵) = (𝐹‘𝐵) ↔ 𝐵𝐹(𝐹‘𝐵)))) |
| 6 | 5 | imbi2d 340 | . . . . 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 6678 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ∃!𝑥 𝐵𝐹𝑥) | |
| 8 | tz6.12c 6928 | . . . . . 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 3558 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = (𝐹‘𝐵) ↔ 𝐵𝐹(𝐹‘𝐵))) |
| 11 | 1, 10 | mpbii 233 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵𝐹(𝐹‘𝐵)) |
| 12 | breq2 5147 | . . 3 ⊢ ((𝐹‘𝐵) = 𝐶 → (𝐵𝐹(𝐹‘𝐵) ↔ 𝐵𝐹𝐶)) | |
| 13 | 11, 12 | syl5ibcom 245 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = 𝐶 → 𝐵𝐹𝐶)) |
| 14 | fnfun 6668 | . . . 4 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹) | |
| 15 | funbrfv 6957 | . . . 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 212 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = 𝐶 ↔ 𝐵𝐹𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃!weu 2568 class class class wbr 5143 Fun wfun 6555 Fn wfn 6556 ‘cfv 6561 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 |
| This theorem is referenced by: fnopfvb 6960 funbrfvb 6962 fnbrfvb2 6964 dffn5 6967 feqmptdf 6979 fnsnfv 6988 fndmdif 7062 dffo4 7123 dff13 7275 isomin 7357 isoini 7358 br1steqg 8036 br2ndeqg 8037 1stconst 8125 2ndconst 8126 fsplit 8142 seqomlem3 8492 seqomlem4 8493 nqerrel 10972 imasleval 17586 znleval 21573 scutun12 27855 madeval2 27892 axcontlem5 28983 elnlfn 31947 adjbd1o 32104 fcoinvbr 32618 fv1stcnv 35777 fv2ndcnv 35778 fvbigcup 35903 fvsingle 35921 imageval 35931 brfullfun 35949 bj-mptval 37118 unccur 37610 poimirlem2 37629 poimirlem23 37650 pw2f1ocnv 43049 tfsconcat0i 43358 tfsconcatrev 43361 brcoffn 44043 funressnfv 47055 fnbrafvb 47166 |
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