funbrafv - Mathbox for Alexander van der Vekens

	Mathbox for Alexander van der Vekens		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > funbrafv		Structured version Visualization version GIF version

Theorem funbrafv 47197

Description: The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6870. (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 6498	. . 3 ⊢ (Fun 𝐹 → Rel 𝐹)
2		releldm 5883	. . . . . . . 8 ⊢ ((Rel 𝐹 ∧ 𝐴𝐹𝐵) → 𝐴 ∈ dom 𝐹)
3		funbrafvb 47195	. . . . . . . . . 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 1541 ∈ wcel 2111 class class class wbr 5089 dom cdm 5614 Rel wrel 5619 Fun wfun 6475 '''cafv 47156

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5232 ax-nul 5242 ax-pr 5368

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4281 df-if 4473 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-int 4896 df-br 5090 df-opab 5152 df-id 5509 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-res 5626 df-iota 6437 df-fun 6483 df-fn 6484 df-fv 6489 df-aiota 47124 df-dfat 47158 df-afv 47159

This theorem is referenced by: afvelima 47206

W3C validator