Theorem ofrfvalg 7417

Description: Value of a relation applied to two functions. Originally part of ofrfval 7419, this version assumes the functions are sets rather than their domains, avoiding ax-rep 5159. (Contributed by SN, 5-Aug-2024.)

Hypotheses

Ref	Expression
ofrfvalg.1	⊢ (𝜑 → 𝐹 Fn 𝐴)
ofrfvalg.2	⊢ (𝜑 → 𝐺 Fn 𝐵)
ofrfvalg.3	⊢ (𝜑 → 𝐹 ∈ 𝑉)
ofrfvalg.4	⊢ (𝜑 → 𝐺 ∈ 𝑊)
ofrfvalg.5	⊢ (𝐴 ∩ 𝐵) = 𝑆
ofrfvalg.6	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐶)
ofrfvalg.7	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = 𝐷)

Assertion

Ref	Expression
ofrfvalg	⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑥 ∈ 𝑆 𝐶𝑅𝐷))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐺   𝜑,𝑥   𝑥,𝑆   𝑥,𝑅

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝐷(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem ofrfvalg

Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ofrfvalg.3	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
2		ofrfvalg.4	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑊)
3		dmeq 5748	. . . . . 6 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
4		dmeq 5748	. . . . . 6 ⊢ (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺)
5	3, 4	ineqan12d 4121	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (dom 𝑓 ∩ dom 𝑔) = (dom 𝐹 ∩ dom 𝐺))
6		fveq1 6661	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
7		fveq1 6661	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑥) = (𝐺‘𝑥))
8	6, 7	breqan12d 5051	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓‘𝑥)𝑅(𝑔‘𝑥) ↔ (𝐹‘𝑥)𝑅(𝐺‘𝑥)))
9	5, 8	raleqbidv 3319	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓‘𝑥)𝑅(𝑔‘𝑥) ↔ ∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹‘𝑥)𝑅(𝐺‘𝑥)))
10		df-ofr 7411	. . . 4 ⊢ ∘r 𝑅 = {⟨𝑓, 𝑔⟩ ∣ ∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓‘𝑥)𝑅(𝑔‘𝑥)}
11	9, 10	brabga 5394	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹‘𝑥)𝑅(𝐺‘𝑥)))
12	1, 2, 11	syl2anc 587	. 2 ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹‘𝑥)𝑅(𝐺‘𝑥)))
13		ofrfvalg.1	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn 𝐴)
14	13	fndmd 6442	. . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴)
15		ofrfvalg.2	. . . . . 6 ⊢ (𝜑 → 𝐺 Fn 𝐵)
16	15	fndmd 6442	. . . . 5 ⊢ (𝜑 → dom 𝐺 = 𝐵)
17	14, 16	ineq12d 4120	. . . 4 ⊢ (𝜑 → (dom 𝐹 ∩ dom 𝐺) = (𝐴 ∩ 𝐵))
18		ofrfvalg.5	. . . 4 ⊢ (𝐴 ∩ 𝐵) = 𝑆
19	17, 18	eqtrdi 2809	. . 3 ⊢ (𝜑 → (dom 𝐹 ∩ dom 𝐺) = 𝑆)
20	19	raleqdv 3329	. 2 ⊢ (𝜑 → (∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹‘𝑥)𝑅(𝐺‘𝑥) ↔ ∀𝑥 ∈ 𝑆 (𝐹‘𝑥)𝑅(𝐺‘𝑥)))
21		inss1 4135	. . . . . . 7 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
22	18, 21	eqsstrri 3929	. . . . . 6 ⊢ 𝑆 ⊆ 𝐴
23	22	sseli 3890	. . . . 5 ⊢ (𝑥 ∈ 𝑆 → 𝑥 ∈ 𝐴)
24		ofrfvalg.6	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐶)
25	23, 24	sylan2 595	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝐹‘𝑥) = 𝐶)
26		inss2 4136	. . . . . . 7 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
27	18, 26	eqsstrri 3929	. . . . . 6 ⊢ 𝑆 ⊆ 𝐵
28	27	sseli 3890	. . . . 5 ⊢ (𝑥 ∈ 𝑆 → 𝑥 ∈ 𝐵)
29		ofrfvalg.7	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = 𝐷)
30	28, 29	sylan2 595	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝐺‘𝑥) = 𝐷)
31	25, 30	breq12d 5048	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((𝐹‘𝑥)𝑅(𝐺‘𝑥) ↔ 𝐶𝑅𝐷))
32	31	ralbidva 3125	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝑆 (𝐹‘𝑥)𝑅(𝐺‘𝑥) ↔ ∀𝑥 ∈ 𝑆 𝐶𝑅𝐷))
33	12, 20, 32	3bitrd 308	1 ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑥 ∈ 𝑆 𝐶𝑅𝐷))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070   ∩ cin 3859   class class class wbr 5035  dom cdm 5527   Fn wfn 6334  ‘cfv 6339   ∘_r cofr 7409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rab 3079  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5036  df-opab 5098  df-dm 5537  df-iota 6298  df-fn 6342  df-fv 6347  df-ofr 7411

This theorem is referenced by:  ofrfval  7419  pwsle  16828  pwsleval  16829  psrbaglesupp  20691  psrbaglefi  20699