funbrafv - Mathbox for Alexander van der Vekens

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Theorem funbrafv 47154

Description: The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6932. (Contributed by Alexander van der Vekens, 25-May-2017.)

**Assertion**
Ref	Expression
funbrafv	⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵))

**Proof of Theorem funbrafv**
Step	Hyp	Ref	Expression
1		funrel 6558	. . 3 ⊢ (Fun 𝐹 → Rel 𝐹)
2		releldm 5929	. . . . . . . 8 ⊢ ((Rel 𝐹 ∧ 𝐴𝐹𝐵) → 𝐴 ∈ dom 𝐹)
3		funbrafvb 47152	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹'''𝐴) = 𝐵 ↔ 𝐴𝐹𝐵))
4	3	biimprd 248	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵))
5	4	expcom 413	. . . . . . . 8 ⊢ (𝐴 ∈ dom 𝐹 → (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵)))
6	2, 5	syl 17	. . . . . . 7 ⊢ ((Rel 𝐹 ∧ 𝐴𝐹𝐵) → (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵)))
7	6	ex 412	. . . . . 6 ⊢ (Rel 𝐹 → (𝐴𝐹𝐵 → (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵))))
8	7	com14 96	. . . . 5 ⊢ (𝐴𝐹𝐵 → (𝐴𝐹𝐵 → (Fun 𝐹 → (Rel 𝐹 → (𝐹'''𝐴) = 𝐵))))
9	8	pm2.43i 52	. . . 4 ⊢ (𝐴𝐹𝐵 → (Fun 𝐹 → (Rel 𝐹 → (𝐹'''𝐴) = 𝐵)))
10	9	com13 88	. . 3 ⊢ (Rel 𝐹 → (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵)))
11	1, 10	syl 17	. 2 ⊢ (Fun 𝐹 → (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵)))
12	11	pm2.43i 52	1 ⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 class class class wbr 5124 dom cdm 5659 Rel wrel 5664 Fun wfun 6530 '''cafv 47113

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-sep 5271 ax-nul 5281 ax-pr 5407

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3062 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-int 4928 df-br 5125 df-opab 5187 df-id 5553 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-res 5671 df-iota 6489 df-fun 6538 df-fn 6539 df-fv 6544 df-aiota 47081 df-dfat 47115 df-afv 47116

This theorem is referenced by: afvelima 47163

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