funbrafv - Mathbox for Alexander van der Vekens

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Theorem funbrafv 48227

Description: The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6933. (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 6556	. . 3 ⊢ (Fun 𝐹 → Rel 𝐹)
2		releldm 5926	. . . . . . . 8 ⊢ ((Rel 𝐹 ∧ 𝐴𝐹𝐵) → 𝐴 ∈ dom 𝐹)
3		funbrafvb 48225	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹'''𝐴) = 𝐵 ↔ 𝐴𝐹𝐵))
4	3	biimprd 251	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵))
5	4	expcom 419	. . . . . . . 8 ⊢ (𝐴 ∈ dom 𝐹 → (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵)))
6	2, 5	syl 18	. . . . . . 7 ⊢ ((Rel 𝐹 ∧ 𝐴𝐹𝐵) → (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵)))
7	6	ex 418	. . . . . 6 ⊢ (Rel 𝐹 → (𝐴𝐹𝐵 → (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵))))
8	7	com14 97	. . . . 5 ⊢ (𝐴𝐹𝐵 → (𝐴𝐹𝐵 → (Fun 𝐹 → (Rel 𝐹 → (𝐹'''𝐴) = 𝐵))))
9	8	pm2.43i 53	. . . 4 ⊢ (𝐴𝐹𝐵 → (Fun 𝐹 → (Rel 𝐹 → (𝐹'''𝐴) = 𝐵)))
10	9	com13 89	. . 3 ⊢ (Rel 𝐹 → (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵)))
11	1, 10	syl 18	. 2 ⊢ (Fun 𝐹 → (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵)))
12	11	pm2.43i 53	1 ⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹'''𝐴) = 𝐵))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 class class class wbr 5103 dom cdm 5651 Rel wrel 5656 Fun wfun 6532 '''cafv 48186

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-sep 5249 ax-nul 5260 ax-pr 5391

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-br 5104 df-opab 5168 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-res 5663 df-iota 6494 df-fun 6540 df-fn 6541 df-fv 6546 df-aiota 48154 df-dfat 48188 df-afv 48189

This theorem is used by: afvelima 48236

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