funbrafv - Mathbox for Alexander van der Vekens

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Theorem funbrafv 47073

Description: The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6971. (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 6595	. . 3 ⊢ (Fun 𝐹 → Rel 𝐹)
2		releldm 5969	. . . . . . . 8 ⊢ ((Rel 𝐹 ∧ 𝐴𝐹𝐵) → 𝐴 ∈ dom 𝐹)
3		funbrafvb 47071	. . . . . . . . . 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 1537 ∈ wcel 2108 class class class wbr 5166 dom cdm 5700 Rel wrel 5705 Fun wfun 6567 '''cafv 47032

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-br 5167 df-opab 5229 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-res 5712 df-iota 6525 df-fun 6575 df-fn 6576 df-fv 6581 df-aiota 47000 df-dfat 47034 df-afv 47035

This theorem is referenced by: afvelima 47082

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