Theorem ofrfval2 7676

Description: The function relation acting on maps. (Contributed by Mario Carneiro, 20-Jul-2014.)

Hypotheses

Ref	Expression
offval2.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
offval2.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)
offval2.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑋)
offval2.4	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))
offval2.5	⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐶))

Assertion

Ref	Expression
ofrfval2	⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑥 ∈ 𝐴 𝐵𝑅𝐶))

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝑥,𝑅

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝐹(𝑥)   𝐺(𝑥)   𝑉(𝑥)   𝑊(𝑥)   𝑋(𝑥)

Proof of Theorem ofrfval2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		offval2.2	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)
2	1	ralrimiva 3153	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊)
3		eqid 2761	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
4	3	fnmpt 6656	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑊 → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
5	2, 4	syl 17	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
6		offval2.4	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))
7	6	fneq1d 6609	. . . 4 ⊢ (𝜑 → (𝐹 Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴))
8	5, 7	mpbird 259	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
9		offval2.3	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑋)
10	9	ralrimiva 3153	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝑋)
11		eqid 2761	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶)
12	11	fnmpt 6656	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝑋 → (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴)
13	10, 12	syl 17	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴)
14		offval2.5	. . . . 5 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐶))
15	14	fneq1d 6609	. . . 4 ⊢ (𝜑 → (𝐺 Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴))
16	13, 15	mpbird 259	. . 3 ⊢ (𝜑 → 𝐺 Fn 𝐴)
17		offval2.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
18		inidm 4176	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
19	6	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))
20	19	fveq1d 6864	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦))
21	14	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐶))
22	21	fveq1d 6864	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐺‘𝑦) = ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦))
23	8, 16, 17, 17, 18, 20, 22	ofrfval 7665	. 2 ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑦 ∈ 𝐴 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)𝑅((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦)))
24		nffvmpt1 6873	. . . . 5 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)
25		nfcv 2923	. . . . 5 ⊢ Ⅎ𝑥𝑅
26		nffvmpt1 6873	. . . . 5 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦)
27	24, 25, 26	nfbr 5144	. . . 4 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)𝑅((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦)
28		nfv 1933	. . . 4 ⊢ Ⅎ𝑦((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)𝑅((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥)
29		fveq2 6862	. . . . 5 ⊢ (𝑦 = 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
30		fveq2 6862	. . . . 5 ⊢ (𝑦 = 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) = ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥))
31	29, 30	breq12d 5110	. . . 4 ⊢ (𝑦 = 𝑥 → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)𝑅((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)𝑅((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥)))
32	27, 28, 31	cbvralw 3303	. . 3 ⊢ (∀𝑦 ∈ 𝐴 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)𝑅((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) ↔ ∀𝑥 ∈ 𝐴 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)𝑅((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥))
33		simpr 488	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
34	3	fvmpt2 6982	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑊) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
35	33, 1, 34	syl2anc 593	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
36	11	fvmpt2 6982	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝑋) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶)
37	33, 9, 36	syl2anc 593	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) = 𝐶)
38	35, 37	breq12d 5110	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)𝑅((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) ↔ 𝐵𝑅𝐶))
39	38	ralbidva 3182	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)𝑅((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑥) ↔ ∀𝑥 ∈ 𝐴 𝐵𝑅𝐶))
40	32, 39	bitrid 285	. 2 ⊢ (𝜑 → (∀𝑦 ∈ 𝐴 ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)𝑅((𝑥 ∈ 𝐴 ↦ 𝐶)‘𝑦) ↔ ∀𝑥 ∈ 𝐴 𝐵𝑅𝐶))
41	23, 40	bitrd 281	1 ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑥 ∈ 𝐴 𝐵𝑅𝐶))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075   class class class wbr 5097   ↦ cmpt 5178   Fn wfn 6511  ‘cfv 6516   ∘_r cofr 7654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-ofr 7656

This theorem is referenced by:  gsumbagdiaglem  21971  mplmonmul  22077  coe1mul2lem1  22318  itg2const  25790  itg2const2  25791  itg2uba  25793  itg2mulclem  25796  itg2splitlem  25798  itg2split  25799  itg2monolem1  25800  itg2gt0  25810  itg2cnlem1  25811  itg2cnlem2  25812  iblss  25855  i1fibl  25858  itgitg1  25859  itgle  25860  ibladdlem  25870  iblabs  25879  iblabsr  25880  iblmulc2  25881  bddmulibl  25889  bddiblnc  25892  mplmulmvr  33797  psrmonmul  33808  itg2addnclem  38131  itg2addnclem3  38133  itg2addnc  38134  itg2gt0cn  38135  ibladdnclem  38136  iblabsnc  38144  iblmulc2nc  38145  ftc1anclem4  38156  ftc1anclem5  38157  ftc1anclem6  38158  ftc1anclem7  38159  ftc1anclem8  38160  ftc1anc  38161