Theorem caofid0r 7754

Description: Transfer a right identity law to the function operation. (Contributed by NM, 21-Oct-2014.)

Hypotheses

Ref	Expression
caofref.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
caofref.2	⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
caofid0.3	⊢ (𝜑 → 𝐵 ∈ 𝑊)
caofid0r.5	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥𝑅𝐵) = 𝑥)

Assertion

Ref	Expression
caofid0r	⊢ (𝜑 → (𝐹 ∘f 𝑅(𝐴 × {𝐵})) = 𝐹)

Distinct variable groups:   𝑥,𝐵   𝑥,𝐹   𝜑,𝑥   𝑥,𝑅   𝑥,𝑆

Allowed substitution hints:   𝐴(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem caofid0r

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		caofref.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		caofref.2	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
3	2	ffnd 6735	. 2 ⊢ (𝜑 → 𝐹 Fn 𝐴)
4		caofid0.3	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
5		fnconstg 6795	. . 3 ⊢ (𝐵 ∈ 𝑊 → (𝐴 × {𝐵}) Fn 𝐴)
6	4, 5	syl 18	. 2 ⊢ (𝜑 → (𝐴 × {𝐵}) Fn 𝐴)
7		eqidd 2764	. 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) = (𝐹‘𝑤))
8		fvconst2g 7234	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑤 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
9	4, 8	sylan 592	. 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑤) = 𝐵)
10		caofid0r.5	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥𝑅𝐵) = 𝑥)
11	10	ralrimiva 3158	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (𝑥𝑅𝐵) = 𝑥)
12	2	ffvelcdmda 7111	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆)
13		oveq1 7453	. . . . 5 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑅𝐵) = ((𝐹‘𝑤)𝑅𝐵))
14		id 23	. . . . 5 ⊢ (𝑥 = (𝐹‘𝑤) → 𝑥 = (𝐹‘𝑤))
15	13, 14	eqeq12d 2779	. . . 4 ⊢ (𝑥 = (𝐹‘𝑤) → ((𝑥𝑅𝐵) = 𝑥 ↔ ((𝐹‘𝑤)𝑅𝐵) = (𝐹‘𝑤)))
16	15	rspccva 3579	. . 3 ⊢ ((∀𝑥 ∈ 𝑆 (𝑥𝑅𝐵) = 𝑥 ∧ (𝐹‘𝑤) ∈ 𝑆) → ((𝐹‘𝑤)𝑅𝐵) = (𝐹‘𝑤))
17	11, 12, 16	syl2an2r 698	. 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤)𝑅𝐵) = (𝐹‘𝑤))
18	1, 3, 6, 3, 7, 9, 17	offveq 7746	1 ⊢ (𝜑 → (𝐹 ∘f 𝑅(𝐴 × {𝐵})) = 𝐹)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  {csn 4588   × cxp 5654   Fn wfn 6556  ⟶wf 6557  ‘cfv 6561  (class class class)co 7446   ∘_f cof 7718

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pr 5396

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-of 7720

This theorem is used by:  mndvrid  19027  psrlidm  22302  psdmul  22520  lfl1sc  40344