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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > funbrafv22b | Structured version Visualization version GIF version |
Description: Equivalence of function value and binary relation, analogous to funbrfvb 6961. (Contributed by AV, 6-Sep-2022.) |
Ref | Expression |
---|---|
funbrafv22b | ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹''''𝐴) = 𝐵 ↔ 𝐴𝐹𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funfn 6597 | . 2 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹) | |
2 | fnbrafv2b 47197 | . 2 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹''''𝐴) = 𝐵 ↔ 𝐴𝐹𝐵)) | |
3 | 1, 2 | sylanb 581 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹''''𝐴) = 𝐵 ↔ 𝐴𝐹𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 class class class wbr 5147 dom cdm 5688 Fun wfun 6556 Fn wfn 6557 ''''cafv2 47157 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-res 5700 df-iota 6515 df-fun 6564 df-fn 6565 df-dfat 47068 df-afv2 47158 |
This theorem is referenced by: (None) |
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