funbrfv - Intuitionistic Logic Explorer

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Theorem funbrfv 5155

Description: The second argument of a binary relation on a function is the function's value. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)

**Assertion**
Ref	Expression
funbrfv	⊢ (Fun 𝐹 → (A𝐹B → (𝐹‘A) = B))

Proof of Theorem funbrfv Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funrel 4862	. . . 4 ⊢ (Fun 𝐹 → Rel 𝐹)
2		brrelex2 4326	. . . 4 ⊢ ((Rel 𝐹 ∧ A𝐹B) → B ∈ V)
3	1, 2	sylan 267	. . 3 ⊢ ((Fun 𝐹 ∧ A𝐹B) → B ∈ V)
4		breq2 3759	. . . . . 6 ⊢ (y = B → (A𝐹y ↔ A𝐹B))
5	4	anbi2d 437	. . . . 5 ⊢ (y = B → ((Fun 𝐹 ∧ A𝐹y) ↔ (Fun 𝐹 ∧ A𝐹B)))
6		eqeq2 2046	. . . . 5 ⊢ (y = B → ((𝐹‘A) = y ↔ (𝐹‘A) = B))
7	5, 6	imbi12d 223	. . . 4 ⊢ (y = B → (((Fun 𝐹 ∧ A𝐹y) → (𝐹‘A) = y) ↔ ((Fun 𝐹 ∧ A𝐹B) → (𝐹‘A) = B)))
8		funeu 4869	. . . . . 6 ⊢ ((Fun 𝐹 ∧ A𝐹y) → ∃!y A𝐹y)
9		tz6.12-1 5143	. . . . . 6 ⊢ ((A𝐹y ∧ ∃!y A𝐹y) → (𝐹‘A) = y)
10	8, 9	sylan2 270	. . . . 5 ⊢ ((A𝐹y ∧ (Fun 𝐹 ∧ A𝐹y)) → (𝐹‘A) = y)
11	10	anabss7 517	. . . 4 ⊢ ((Fun 𝐹 ∧ A𝐹y) → (𝐹‘A) = y)
12	7, 11	vtoclg 2607	. . 3 ⊢ (B ∈ V → ((Fun 𝐹 ∧ A𝐹B) → (𝐹‘A) = B))
13	3, 12	mpcom 32	. 2 ⊢ ((Fun 𝐹 ∧ A𝐹B) → (𝐹‘A) = B)
14	13	ex 108	1 ⊢ (Fun 𝐹 → (A𝐹B → (𝐹‘A) = B))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 97 = wceq 1242 ∈ wcel 1390 ∃!weu 1897 Vcvv 2551 class class class wbr 3755 Rel wrel 4293 Fun wfun 4839 ‘cfv 4845

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bnd 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935

This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-sbc 2759 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-iota 4810 df-fun 4847 df-fv 4853

This theorem is referenced by: funopfv 5156 fnbrfvb 5157 fvelima 5168 fvi 5173 fmptco 5273 fliftfun 5379 fliftval 5383 tfrlem5 5871

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